(in formats from the Mathematica program, from Wolfram Research, Inc.)

SeriesAtLevelR  =

Underoverscript[∑, i = 1, arg3] (-i + n + r + x) Eulerian(n, i - 1) n + r

(employing the "add-in" <<DiscreteMath`Combinatorica`)

FullSimplify[Table [SeriesAtLevelR, {r,-1,5},{x,x,x},{n,1,14}]]//TableForm

see also these formulas expanded and these formulas factored or a chart of initial numerical values

 

r=-1

n=1    1    ones or    1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).
n=2    2 x - 1      A005408  The odd numbers.
n=3    3 (x - 1) x + 1  A003215 Hex (or centered hexagonal) numbers (crystal ball sequence for hexagonal lattice).
n=4    2 x (x (2 x - 3) + 2) - 1   A005917  Rhombic dodecahedral numbers
n=5    5 (x - 1) x ((x - 1) x + 1) + 1   A022521  Nexus numbers for the power of 5.
n=6    (2 x - 1) ((x - 1) x + 1) (3 (x - 1) x + 1)      A022522  Nexus numbers for the power of 6.
n=7    7 (x - 1) x ((x - 1) x + 1)^2 + 1     A022523  Nexus numbers for the power of 7.
n=8    (2 x - 1) (2 (x - 1) x + 1) (2 (x - 1) x ((x - 1) x + 2) + 1)      A022524  Nexus numbers for the power of 8.
n=9    (3 (x - 1) x + 1) (3 (x - 1) x ((x - 1) x + 1) ((x - 1) x + 2) + 1)      A022525  Nexus numbers for the power of 9.
n=10    (2 x - 1) (5 (x - 1) x ((x - 1) x + 1) + 1) ((x - 1) x ((x - 1) x + 3) + 1)       A022526  Nexus numbers for the power of 10.
n=11    11 (x - 1) x ((x - 1) x + 1) ((x - 1) x ((x - 1) x + 1) ((x - 1) x + 3) + 1) + 1     A022527  Nexus numbers for the power of 11.
n=12    (2 x - 1) ((x - 1) x + 1) (2 (x - 1) x + 1) (3 (x - 1) x + 1) ((x - 1) x ((x - 1) x + 4) + 1)     A022528  Nexus numbers for the power of 12.
n=13    13 (x - 1) x ((x - 1) x ((x - 1) x ((x - 1) x + 5) + 3) + 1) ((x - 1) x + 1)^2 + 1     A022529  Nexus numbers for the power of 13.
n=14    (2 x - 1) (7 (x - 1) x ((x - 1) x + 1)^2 + 1) ((x - 1) x ((x - 1) x + 1) ((x - 1) x + 5) + 1)      A022530  Nexus numbers for the power of 14.
 

r=0

n=1    x      Integers or the power of 1.
n=2    x^2   A000290  The squares.
n=3    x^3   A000578  The cubes.
n=4    x^4   A000583  The fourth power.  "Figurate numbers based on 4-dimensional regular convex polytope
              called the 4-measure polytope, 4-hypercube or tessaract with
              Schlafli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004"
n=5    x^5  A000584  The fifth power.
n=6    x^6  A001014  The sixth power.  Numbers both square and cubic - pdg(AT)worldofnumbers.com.
n=7    x^7   A001015  The seventh power.
n=8    x^8   A001016  The eighth power.
n=9    x^9   A001017   The ninth power.
n=10   x^10  A008454  The tenth power.
n=11   x^11   A008455  The eleventh power.
n=12   x^12   A008456  The twelfth power.
n=13   x^13   A010801  The 13th power.
n=14   x^14  A010802  The 14th power.
 

r=1

n=1    1/2 x (x + 1)     A000217    3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
n=2    1/6 x (x + 1) (2 x + 1)   A000330  Square pyramidal numbers. 
n=3    1/4 x^2 (x + 1)^2   A000537  Sum of first n cubes; or n-th triangular number squared.
n=4    1/30 x (x + 1) (2 x + 1) (3 x (x + 1) - 1)   A000538  Sum of fourth powers: 0^4+1^4+...+n^4.
n=5    1/12 x^2 (x + 1)^2 (2 x (x + 1) - 1)  A000539  Sum of 5th powers: 1^5 + 2^5 + ... + n^5.
n=6    1/42 (6 x^7 + 21 x^6 + 21 x^5 - 7 x^3 + x)  A000540  Sum of 6th powers: 1^6 + 2^6 + ... + n^6.
n=7    1/24 (3 x^8 + 12 x^7 + 14 x^6 - 7 x^4 + 2 x^2)  A000541  Sum of 7th powers: 1^7 + 2^7 + ... + n^7.
n=8    1/90 x (x^2 ((5 x^2 (x (2 x + 9) + 12) - 42) x^2 + 20) - 3)  A000542  Sum of 8th powers: 1^8 + 2^8 + ... + n^8.
n=9    1/20 x^2 (x^2 ((x^2 (2 x (x + 5) + 15) - 14) x^2 + 10) - 3)  A007487  Sum of 9th powers.
n=10   1/66 x (6 x^10 + 33 x^9 + 55 x^8 - 66 x^6 + 66 x^4 - 33 x^2 + 5)   A023002  Sum of 10th powers.
n=11   1/24 x^2 ((x^2 ((2 x^2 (x (x + 6) + 11) - 33) x^2 + 44) - 33) x^2 + 10)  Sum of 11th powers.
n=12   x^13/13 + x^12/2 + x^11 - (11 x^9)/6 + (22 x^7)/7 - (33 x^5)/10 + (5 x^3)/3 - (691 x)/2730  Sum of 12th powers.
n=13   1/420 x^2 (x^2 ((x^2 ((5 x^2 (6 x (x + 7) + 91) - 1001) x^2 + 2145) - 3003) x^2 + 2275) - 691)  Sum of 13th powers.
n=14   1/90 x ((x^2 ((x^2 (3 (x^2 (x (2 x + 15) + 35) - 91) x^2 + 715) - 1287) x^2 + 1365) - 691) x^2 + 105)  Sum of 14th powers.
 

r=2

n=1    1/6 x (x + 1) (x + 2)  A000292   4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
n=2    1/12 x (x + 1)^2 (x + 2)    A002415  4-dimensional pyramidal numbers: n^2*(n^2-1)/12.

                                     Also number of ways to legally insert two pairs of parentheses into a
                                            string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827)
                                            ways to insert the parentheses, but we must subtract 2(m+1) for illegal
                                            clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2)
                                            for 2 clumps of 2 parentheses, and (m-1)C(m+1,2) for 1 clump of 2
                                            parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.)  E.g. for n=2

                                     there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)),a((b)).
n=3    1/60 x (x + 1) (x + 2) (3 x (x + 2) + 1)   A085437  or  A024166  Sum of (j-i)^3 for 1 <= i < j <= n.
n=4    1/60 x (x + 1)^2 (x + 2) (2 x (x + 2) - 1)
n=5    1/84 x (x + 1) (x + 2) (x (x + 2) - 1) (2 x (x + 2) + 1)
n=6    1/168 x (x + 1)^2 (x + 2) (x (x + 2) - 1) (3 x (x + 2) - 2)
n=7    1/360 x (x + 1) (x + 2) (x (x (5 x^2 (x (x + 6) + 10) - 37) + 6) + 6)
n=8    1/180 x (x + 1)^2 (x + 2) (2 x (x + 2) - 1) ((x - 1) x (x + 2) (x + 3) + 3)
n=9    1/660 x (x + 1) (x + 2) (x^2 + x - 1) (x (x + 3) + 1) (x (x + 2) (6 x (x + 2) - 19) + 25)
n=10   1/264 x (x + 1)^2 (x + 2) (x (x + 2) - 2) (x (x + 2) (2 x^4 + 8 x^3 - 16 x + 15) - 5)
n=11   (x (x + 1) (x + 2) (x (x + 2) (5 x (x + 2) (7 x (x + 2) (2 x (x + 2) (x (x + 2) - 7) + 43) - 396) + 101) + 1382))/10920
n=12    (x (x + 1)^2 (x + 2) (x (x + 2) (x (x + 2) (x (x + 2) (5 x (x + 2) (6 x (x + 2) - 55) + 1178) - 2663) + 2764) - 691))/5460
n=13   (x (x + 1) (x + 2) (x (x + 2) (x (x + 2) (x (x + 2) (3 x (x + 2) (x (x + 2) (2 x (x + 2) - 21) + 105) - 845) + 1009) + 33) - 735))/1260
n=14   1/720 x (x + 1)^2 (x + 2) (x (x + 2) (x (x + 2) (x (x + 2) (3 x (x + 2) (x (x + 2) (x (x + 2) - 13) + 83) - 925) + 1934) - 1890) + 420)
 

r=3

n=1    1/24 x (x + 1) (x + 2) (x + 3)     A000332  5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
n=2    1/120 x (x + 1) (x + 2) (x + 3) (2 x + 3)   A005585  5-dimensional pyramidal numbers: n(n+1) ... (n+3)(2n+3)/5!.
n=3    1/120 x (x + 1) (x + 2) (x + 3) (x (x + 3) + 1)
n=4    1/840 x (x + 1) (x + 2) (x + 3) (2 x + 3) (2 x (x + 3) - 1)
n=5    1/336 x (x + 1) (x + 2) (x + 3) (x (x + 2) - 1) (x (x + 4) + 2)
n=6    (x (x + 1) (x + 2) (x + 3) (2 x + 3) (5 x (x + 3) (x (x + 3) - 2) + 2))/5040
n=7    1/720 x (x + 1) (x + 2) (x + 3) (x (x + 3) (x (x + 2) - 2) (x (x + 4) + 1) + 6)
n=8    (x (x + 1) (x + 2) (x + 3) (2 x + 3) (x (x (x^2 (2 x (x + 9) + 45) - 69) + 36) + 1))/3960
n=9    (x (x + 1) (x + 2) (x + 3) (x (x + 3) (x (x + 3) (2 x (x + 3) (x (x + 3) - 5) + 11) + 28) - 50))/2640
n=10   (x (x + 1) (x + 2) (x + 3) (2 x + 3) (5 x (x + 3) (7 x (x + 3) (2 x (x + 3) (x (x + 3) - 8) + 49) - 342) - 678))/240240
n=11   (x (x + 1) (x + 2) (x + 3) (x (x + 3) (x (x + 3) (x (x + 3) (10 x (x + 3) (x (x + 3) - 9) + 299) - 168) - 1030) + 1382))/21840
n=12   (x (x + 1) (x + 2) (x + 3) (2 x + 3) (x (x + 3) (x (x + 3) (3 x (x + 3) (x (x + 3) (2 x (x + 3) - 25) + 134) - 1004) + 766) + 601))/32760
n=13   (x (x + 1) (x + 2) (x + 3) (x (x + 3) (x (x + 3) (x (x + 3) (3 x (x + 3) (x (x + 3) (x (x + 3) - 14) + 84) - 664) + 86) + 2672) - 2940))/10080
n=14   (x (x + 1) (x + 2) (x + 3) (2 x + 3) (x (x + 3) (x (x + 3) (x (x + 3) (3 x (x + 3) ((x - 3) x (x + 3) (x + 6) + 148) - 1960) + 4254) - 2460) - 3236))/24480
 

r=4

n=1    1/120 x (x + 1) (x + 2) (x + 3) (x + 4)     A000389  6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
n=2    1/360 x (x + 1) (x + 2)^2 (x + 3) (x + 4)   A040977  C(n+5,5)*(n+3)/3; Sequence is n^2*(n^2-1)*(n^2-4)/360 if offset 3.
n=3     1/840 x (x + 1) (x + 2) (x + 3) (x + 4) (x (x + 4) + 2)
n=4    (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (3 x (x + 4) - 1))/5040
n=5    (x (x + 1) (x + 2) (x + 3) (x + 4) (5 x (x + 4) (x (x + 4) + 1) - 24))/15120
n=6    (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x (x + 4) (x (x + 4) - 2) - 1))/5040
n=7    (x (x + 1) (x + 2) (x + 3) (x + 4) (x (x + 4) (x (x + 4) (3 x (x + 4) - 4) - 25) + 48))/23760
n=8    (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x (x + 4) + 1) (2 x (x + 4) (x (x + 4) - 6) + 21))/23760
n=9    (x (x + 1) (x + 2) (x + 3) (x + 4) (x (x + 4) (7 x (x + 4) (6 (x - 1) x (x + 4) (x + 5) - 17) + 2377) - 2904))/720720
n=10   (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x (x + 4) (x (x + 4) (10 x (x + 4) (3 x (x + 4) - 28) + 851) - 72) - 2663))/720720
n=11   (x (x + 1) (x + 2) (x + 3) (x + 4) (x (x + 4) (x (x + 4) (3 x (x + 4) (2 x (x + 4) (x (x + 4) - 10) + 53) + 458) - 2629) + 2208))/196560
n=12   (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x (x + 4) (x (x + 4) (x (x + 4) (3 x (x^3 + 8 x^2 + x - 60) + 265) - 567) - 466) + 2494))/131040
n=13   (x (x + 1) (x + 2) (x + 3) (x + 4) (x (x + 4) (x (x + 4) (3 x (x + 4) (x (x + 4) (x (x + 4) (3 x (x + 4) - 49) + 301) - 419) - 9794) + 36226) - 20496))/514080
n=14   (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x (x + 4) (x (x + 4) (x (x + 4) (x (x + 4) (x (x + 4) (x (x + 4) - 22) + 208) - 950) + 1441) + 3024) - 9066))/73440
 

r=5

n=1    1/720 x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5)      A000579  7th Pascal Triangle Figurate numbers or binomial coefficients C(n,6).
n=2     (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5))/5040   A050486  C(n+6,6)*(2n+7)/7.
n=3    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (3 x (x + 5) + 10))/20160
n=4    (x^2 (x + 1) (x + 2) (x + 3) (x + 4) (x + 5)^2 (2 x + 5))/30240
n=5    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x (x + 5) - 2) (2 x (x + 5) + 9))/60480
n=6    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (x (x + 5) - 3) (3 x (x + 5) + 4))/332640
n=7    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x (x + 5) - 3) (x (x + 5) - 2) (x (x + 5) + 5))/95040
n=8    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (7 x (x + 5) (2 x (x + 5) (x (x + 5) - 5) - 3) + 480))/4324320
n=9    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x (x + 5) (2 x (x + 5) (x (x + 5) (6 x (x + 5) - 25) - 125) + 1585) - 1258))/2882880
n=10   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (x (x + 5) (3 x (x + 5) (2 x (x + 5) (x (x + 5) - 10) + 47) + 455) - 1764))/4324320
n=11   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x (x + 5) (x (x + 5) (3 x (x + 5) (x (x + 5) (x (x + 5) - 10) + 6) + 880) - 3062) + 540))/1572480
n=12   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (x (x + 5) (3 x (x + 5) (x (x + 5) (x (x + 5) (3 x (x + 5) - 50) + 288) - 200) - 10606) + 25680))/13366080
n=13   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x (x + 5) (x (x + 5) (x (x + 5) (x (x + 5) (x (x + 5) (2 x (x + 5) - 35) + 182) + 500) - 7716) + 17770) + 8988))/2056320
n=14   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (7 x (x + 5) (x (x + 5) (x (x + 5) (x (x + 5) (x (x + 5) (x (x + 5) - 25) + 250) - 980) - 1046) + 17610) - 224184))/19535040

FullSimplify[Table [SeriesAtLevelR, {x,1,12},{r,r,r},{n,1,14}]]//TableForm

see also these formulas expanded or a chart of initial numerical values

 

x=1

n=1    1   ones or 1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).
n=2    1
n=3    1
n=4    1
n=5    1
n=6    1
n=7    1
n=8    1
n=9    1
n=10   1
n=11   1
n=12   1
n=13   1
n=14   1

 

x=2

n=1    r + 2     integers or the 2nd Pascal Triangle Figurate numbers or binomial coefficients C(n,1).
n=2    r + 4
n=3    r + 8
n=4    r + 16
n=5    r + 32
n=6    r + 64
n=7    r + 128
n=8    r + 256
n=9    r + 512
n=10   r + 1024
n=11   r + 2048
n=12   r + 4096
n=13   r + 8192
n=14   r + 16384

 

x=3

n=1    1/2 (r + 2) (r + 3)     A000217    3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
n=2    1/2 (r + 3) (r + 6)     A000096
n=3    1/2 (r (r + 17) + 54)   A051936 Truncated triangular numbers: a(n)=n*(n+1)/2-9 or A060533 Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 3 labeled nodes.
n=4    1/2 (r + 6) (r + 27)
n=5    1/2 (r (r + 65) + 486)
n=6    1/2 (r (r + 129) + 1458)
n=7    1/2 (r (r + 257) + 4374)
n=8    1/2 (r + 27) (r + 486)
n=9   1/2 (r (r + 1025) + 39366)
n=10   1/2 (r (r + 2049) + 118098)
n=11   1/2 (r (r + 4097) + 354294)
n=12   1/2 (r (r + 8193) + 1062882)
n=13   1/2 (r (r + 16385) + 3188646)
n=14   1/2 (r (r + 32769) + 9565938)

 

x=4

n=1    1/6 (r + 2) (r + 3) (r + 4)    A000292   4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
n=2    1/6 (r + 3) (r + 4) (r + 8)    A005581   A class of Boolean functions of n variables and rank 2.
n=3    1/6 (r + 4) (r (r + 23) + 96)
n=4    1/6 (r + 8) (r (r + 43) + 192)
n=5    1/6 (r + 12) (r (r + 87) + 512)
n=6    1/6 (r + 8) (r (r + 187) + 3072)
n=7    1/6 (r (r (r + 387) + 13508) + 98304)
n=8    1/6 (r (r (r + 771) + 40136) + 393216)
n=9    1/6 (r (r (r + 1539) + 119636) + 1572864)
n=10   1/6 (r (r (r + 3075) + 357368) + 6291456)
n=11   1/6 (r (r (r + 6147) + 1069028) + 25165824)
n=12   1/6 (r (r (r + 12291) + 3200936) + 100663296)
n=13   1/6 (r (r (r + 24579) + 9590516) + 402653184)
n=14   1/6 (r (r (r + 49155) + 28746968) + 1610612736)

 

x=5

n=1    1/24 (r + 2) (r + 3) (r + 4) (r + 5)     A000332  5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
n=2    1/24 (r + 3) (r + 4) (r + 5) (r + 10)    A005582
n=3    1/24 (r + 4) (r + 5) (r (r + 29) + 150)
n=4    1/24 (r + 5) (r + 10) (r (r + 55) + 300)
n=5    1/24 (r (r (r (r + 134) + 3311) + 27754) + 75000)
n=6    1/24 (r + 10) (r (r (r + 252) + 7007) + 37500)
n=7    1/24 (r (r (r (r + 518) + 27791) + 420490) + 1875000)
n=8    1/24 (r + 10) (r (r (r + 1020) + 71615) + 937500)
n=9    1/24 (r (r (r (r + 2054) + 242351) + 6531754) + 46875000)
n=10   1/24 (r (r (r (r + 4102) + 720887) + 25882610) + 234375000)
n=11   1/24 (r (r (r (r + 8198) + 2150351) + 102805450) + 1171875000)
n=12   1/24 (r (r (r (r + 16390) + 6426455) + 409063250) + 5859375000)
n=13   1/24 (r (r (r (r + 32774) + 19230191) + 1629810154) + 29296875000)
n=14   1/24 (r (r (r (r + 65542) + 57592247) + 6499977650) + 146484375000)

 

x=6

n=1    1/120 (r + 2) (r + 3) (r + 4) (r + 5) (r + 6)     A000389  6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
n=2    1/120 (r + 3) (r + 4) (r + 5) (r + 6) (r + 12)    A005583  Coefficients of Chebyshev polynomials
n=3    1/120 (r + 4) (r + 5) (r + 6) (r + 8) (r + 27)
n=4    1/120 (r + 5) (r + 6) (r + 12) (r (r + 67) + 432)
n=5    1/120 (r + 6) (r (r (r (r + 164) + 4871) + 48604) + 155520)
n=6    1/120 (r + 12) (r (r (r (r + 318) + 12719) + 140442) + 466560)
n=7    1/120 (r (r (r (r (r + 650) + 47615) + 1121350) + 10449384) + 33592320)
n=8    1/120 (r + 12) (r (r (r (r + 1278) + 123599) + 2856762) + 16796160)
n=9    1/120 (r (r (r (r (r + 2570) + 409055) + 16937830) + 250906344) + 1209323520)
n=10   1/120 (r + 12) (r (r (r (r + 5118) + 1150319) + 52710042) + 604661760)
n=11   1/120 (r (r (r (r (r + 10250) + 3604415) + 262399750) + 6118180584) + 43535646720)
n=12   1/120 (r (r (r (r (r + 20490) + 10751735) + 1038744750) + 30324888504) + 261213880320)
n=13   1/120 (r (r (r (r (r + 40970) + 32132255) + 4122641830) + 150574925544) + 1567283281920)
n=14   1/120 (r (r (r (r (r + 81930) + 96150935) + 16394006670) + 748719812664) + 9403699691520)

 

x=7

n=1    1/720 (r + 2) (r + 3) (r + 4) (r + 5) (r + 6) (r + 7)     A000579  7th Pascal Triangle Figurate numbers or binomial coefficients C(n,6).
n=2    1/720 (r + 3) (r + 4) (r + 5) (r + 6) (r + 7) (r + 14)    A005584  Coefficients of Chebyshev polynomials.
n=3    1/720 (r + 4) (r + 5) (r + 6) (r + 7) (r (r + 41) + 294)
n=4    1/720 (r + 5) (r + 6) (r + 7) (r + 14) (r (r + 79) + 588)
n=5    1/720 (r + 6) (r + 7) (r (r (r (r + 194) + 6731) + 77914) + 288120)
n=6    1/720 (r + 7) (r + 14) (r (r (r (r + 378) + 17759) + 226422) + 864360)
n=7    1/720 (r (r (r (r (r (r + 783) + 73375) + 2386845) + 34783624) + 234023292) + 592950960)
n=8    1/720 (r + 14) (r (r (r (r (r + 1537) + 190757) + 6428687) + 76458546) + 296475480)
n=9    1/720 (r (r (r (r (r (r + 3087) + 621295) + 35107965) + 804146104) + 8025597468) + 29054597040)
n=10   1/720 (r + 14) (r (r (r (r (r + 6145) + 1746965) + 112215695) + 2341886274) + 14527298520)
n=11   1/720 (r (r (r (r (r (r + 12303) + 5437375) + 535633245) + 19147147624) + 279830819772) + 1423675254960)
n=12   1/720 (r + 14) (r (r (r (r (r + 24577) + 15844997) + 1887955727) + 67675647186) + 711837627480)
n=13   1/720 (r (r (r (r (r (r + 49167) + 48321295) + 8341762365) + 464140900504) + 9859547101788) + 69760087493040)
n=14   1/720 (r (r (r (r (r (r + 98319) + 144472195) + 33076630005) + 2295485684404) + 58684751577396) + 488320612451280)

Expand[Table [SeriesAtLevelR, {n,n,n},{r,-1,3},{x,1,12}]]//TableForm

see a chart of initial numerical values

  r=-1 r=0 r=1 r=2 r=3
x=1

Underoverscript[∑, i = 1, arg3] (n - i) Eulerian(n, i - 1) n - 1

ones

Underoverscript[∑, i = 1, arg3] (-i + n + 1) Eulerian(n, i - 1) n

ones

Underoverscript[∑, i = 1, arg3] (-i + n + 2) Eulerian(n, i - 1) n + 1

ones

Underoverscript[∑, i = 1, arg3] (-i + n + 3) Eulerian(n, i - 1) n + 2

ones

Underoverscript[∑, i = 1, arg3] (-i + n + 4) Eulerian(n, i - 1) n + 3

ones
x=2

Underoverscript[∑, i = 1, arg3] (-i + n + 1) Eulerian(n, i - 1) n - 1

A000225 2^n - 1.  Sometimes called Mersenne numbers.

Underoverscript[∑, i = 1, arg3] (-i + n + 2) Eulerian(n, i - 1) n  A008776 Pisot sequences* E(2,6), L(2,6), P(2,6), T(2,6).

 or Powers of 2.

Underoverscript[∑, i = 1, arg3] (-i + n + 3) Eulerian(n, i - 1) n + 1 A000051  Pisot sequence* L(2,3)  or 2^n + 1.

Underoverscript[∑, i = 1, arg3] (-i + n + 4) Eulerian(n, i - 1) n + 2 A052548   2^n+2.  Recurrence: {a(0)=3,a(1)=4,

-2*a(n)+a(n+1)+2}

Underoverscript[∑, i = 1, arg3] (-i + n + 5) Eulerian(n, i - 1) n + 3 A062709  2^n+3.  a(n) = 2a(n-1)-3.
x=3

Underoverscript[∑, i = 1, arg3] (-i + n + 2) Eulerian(n, i - 1) n - 1

A001047 3^n - 2^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 3) Eulerian(n, i - 1) n A000244  Pisot sequences* E(1,3), L(1,3), P(1,3), T(1,3) or Powers of 3.

Underoverscript[∑, i = 1, arg3] (-i + n + 4) Eulerian(n, i - 1) n + 1 A001550  1^n + 2^n + 3^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 5) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 6) Eulerian(n, i - 1) n + 3 A087719

"Least number m such that the number of numbers k<=m with
              k>spf(k)^n exceeds the number of numbers with k<=spf(k)^n. 
m<a(n): #{k: k>spf(k)^n & 1<=k<=m} <= m/2;
m>=a(n): #{k: k>spf(k)^n & 1<=k<=m} > m/2."

a(n) = 3^n + 3*2^n + 6.
x=4

Underoverscript[∑, i = 1, arg3] (-i + n + 3) Eulerian(n, i - 1) n - 1

A005061  4^n - 3^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 4) Eulerian(n, i - 1) n A000302  Pisot sequences* E(1,4), L(1,4), P(1,4), T(1,4) or Powers of 4.

Underoverscript[∑, i = 1, arg3] (-i + n + 5) Eulerian(n, i - 1) n + 1 A001551  1^n + 2^n + 3^n + 4^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 6) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 7) Eulerian(n, i - 1) n + 3

 
x=5

Underoverscript[∑, i = 1, arg3] (-i + n + 4) Eulerian(n, i - 1) n - 1

A005060  5^n - 4^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 5) Eulerian(n, i - 1) n A000351 Pisot sequences* E(1,5), L(1,5), P(1,5), T(1,5) or Powers of 5.

 

Underoverscript[∑, i = 1, arg3] (-i + n + 6) Eulerian(n, i - 1) n + 1 A001552  1^n + 2^n + ... + 5^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 7) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 8) Eulerian(n, i - 1) n + 3
x=6

Underoverscript[∑, i = 1, arg3] (-i + n + 5) Eulerian(n, i - 1) n - 1

A005062  6^n - 5^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 6) Eulerian(n, i - 1) n A000400 Pisot sequences* E(1,6), L(1,6), P(1,6), T(1,6)or Powers of 6.

Underoverscript[∑, i = 1, arg3] (-i + n + 7) Eulerian(n, i - 1) n + 1 A001553  1^n + 2^n + ... + 6^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 8) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 9) Eulerian(n, i - 1) n + 3
x=7

Underoverscript[∑, i = 1, arg3] (-i + n + 6) Eulerian(n, i - 1) n - 1

A016169  7^n - 6^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 7) Eulerian(n, i - 1) n A000420  Pisot sequences E(1,7), L(1,7), P(1,7), T(1,7) or Powers of 7.

Underoverscript[∑, i = 1, arg3] (-i + n + 8) Eulerian(n, i - 1) n + 1 A001554  1^n + 2^n + ... + 7^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 9) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 10) Eulerian(n, i - 1) n + 3
x=8

Underoverscript[∑, i = 1, arg3] (-i + n + 7) Eulerian(n, i - 1) n - 1

A016177  8^n - 7^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 8) Eulerian(n, i - 1) n A001018 Pisot sequences* E(1,8), L(1,8), P(1,8), T(1,8) or Powers of 8.

Underoverscript[∑, i = 1, arg3] (-i + n + 9) Eulerian(n, i - 1) n + 1 A001555  1^n + 2^n + ... + 8^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 10) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 11) Eulerian(n, i - 1) n + 3
x=9

Underoverscript[∑, i = 1, arg3] (-i + n + 8) Eulerian(n, i - 1) n - 1

A016185  9^n - 8^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 9) Eulerian(n, i - 1) n A001019 Pisot sequences* E(1,9), L(1,9), P(1,9), T(1,9) or Powers of 9.

Underoverscript[∑, i = 1, arg3] (-i + n + 10) Eulerian(n, i - 1) n + 1 A001556  1^n + 2^n + ... + 9^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 11) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 12) Eulerian(n, i - 1) n + 3
x=10

Underoverscript[∑, i = 1, arg3] (-i + n + 9) Eulerian(n, i - 1) n - 1 A016189  10^n - 9^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 10) Eulerian(n, i - 1) n A011557 Pisot sequences* E(1,10), L(1,10), P(1,10), T(1,10) or Powers of 10.

Underoverscript[∑, i = 1, arg3] (-i + n + 11) Eulerian(n, i - 1) n + 1 A001557  1^n + 2^n + ... + 10^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 12) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 13) Eulerian(n, i - 1) n + 3
x=11

Underoverscript[∑, i = 1, arg3] (-i + n + 10) Eulerian(n, i - 1) n - 1 A016195  11^n - 10^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 11) Eulerian(n, i - 1) n A001020  Pisot sequences* E(1,11), L(1,11), P(1,11), T(1,11) or Powers of 11.

Underoverscript[∑, i = 1, arg3] (-i + n + 12) Eulerian(n, i - 1) n + 1

Underoverscript[∑, i = 1, arg3] (-i + n + 13) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 14) Eulerian(n, i - 1) n + 3
x=12

Underoverscript[∑, i = 1, arg3] (-i + n + 11) Eulerian(n, i - 1) n - 1 A016197  12^n - 11^n.

Underoverscript[∑, i = 1, arg3] (-i + n + 12) Eulerian(n, i - 1) n A001021 Pisot sequences* E(1,12), L(1,12), P(1,12), T(1,12) or Powers of 12.

Underoverscript[∑, i = 1, arg3] (-i + n + 13) Eulerian(n, i - 1) n + 1

Underoverscript[∑, i = 1, arg3] (-i + n + 14) Eulerian(n, i - 1) n + 2

Underoverscript[∑, i = 1, arg3] (-i + n + 15) Eulerian(n, i - 1) n + 3

*Pisot sequence E(x, y): a(0) = x, a(1) = y, a(n) = roundUp(a(n-1)^2/a(n-2)) = [ a(n-1)^2/a(n-2) + 1/2 ].
*Pisot sequence L(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2)).
*Pisot sequence P(x, y): a(0) = x, a(1) = y, a(n) = roundDown(a(n-1)^2/a(n-2)) = ceiling(a(n-1)^2/a(n-2) - 1/2).
*Pisot sequence T(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)) = [ a(n-1)^2/a(n-2) ].

[No Sloane sequence number for .]

Expand[Table [SeriesAtLevelR, {x,1,7},{r,r,r},{n,1,14}]]//TableForm

see also these formulas fully simplified or a chart of initial numerical values

 

x=1

n=1    1    ones or    1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).
n=2    1
n=3    1
n=4    1
n=5    1
n=6    1
n=7    1
n=8    1
n=9    1
n=10   1
n=11   1
n=12   1
n=13   1
n=14   1

 

x=2

n=1    r + 2      integers or the 2nd Pascal Triangle Figurate numbers or binomial coefficients C(n,1).
n=2    r + 4
n=3    r + 8
n=4    r + 16
n=5    r + 32
n=6    r + 64
n=7    r + 128
n=8    r + 256
n=9    r + 512
n=10   r + 1024
n=11   r + 2048
n=12   r + 4096
n=13   r + 8192
n=14   r + 16384

 

x=3

n=1    r^2/2 + (5 r)/2 + 3     A000217    3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
n=2    r^2/2 + (9 r)/2 + 9     A000096
n=3    r^2/2 + (17 r)/2 + 27     A051936 Truncated triangular numbers: a(n)=n*(n+1)/2-9 or A060533 Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 3 labeled nodes.
n=4    r^2/2 + (33 r)/2 + 81
n=5    r^2/2 + (65 r)/2 + 243
n=6    r^2/2 + (129 r)/2 + 729
n=7    r^2/2 + (257 r)/2 + 2187
n=8    r^2/2 + (513 r)/2 + 6561
n=9    r^2/2 + (1025 r)/2 + 19683
n=10   r^2/2 + (2049 r)/2 + 59049
n=11   r^2/2 + (4097 r)/2 + 177147
n=12   r^2/2 + (8193 r)/2 + 531441
n=13   r^2/2 + (16385 r)/2 + 1594323
n=14   r^2/2 + (32769 r)/2 + 4782969

 

x=4

n=1    r^3/6 + (3 r^2)/2 + (13 r)/3 + 4     A000292   4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
n=2    r^3/6 + (5 r^2)/2 + (34 r)/3 + 16     A005581   A class of Boolean functions of n variables and rank 2.
n=3    r^3/6 + (9 r^2)/2 + (94 r)/3 + 64
n=4    r^3/6 + (17 r^2)/2 + (268 r)/3 + 256
n=5    r^3/6 + (33 r^2)/2 + (778 r)/3 + 1024
n=6    r^3/6 + (65 r^2)/2 + (2284 r)/3 + 4096
n=7    r^3/6 + (129 r^2)/2 + (6754 r)/3 + 16384
n=8    r^3/6 + (257 r^2)/2 + (20068 r)/3 + 65536
n=9    r^3/6 + (513 r^2)/2 + (59818 r)/3 + 262144
n=10   r^3/6 + (1025 r^2)/2 + (178684 r)/3 + 1048576
n=11   r^3/6 + (2049 r^2)/2 + (534514 r)/3 + 4194304
n=12   r^3/6 + (4097 r^2)/2 + (1600468 r)/3 + 16777216
n=13   r^3/6 + (8193 r^2)/2 + (4795258 r)/3 + 67108864
n=14   r^3/6 + (16385 r^2)/2 + (14373484 r)/3 + 268435456

 

x=5

n=1     r^4/24 + (7 r^3)/12 + (71 r^2)/24 + (77 r)/12 + 5     A000332  5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
n=2     r^4/24 + (11 r^3)/12 + (167 r^2)/24 + (265 r)/12 + 25     A005582
n=3     r^4/24 + (19 r^3)/12 + (431 r^2)/24 + (965 r)/12 + 125    
n=4     r^4/24 + (35 r^3)/12 + (1175 r^2)/24 + (3625 r)/12 + 625
n=5     r^4/24 + (67 r^3)/12 + (3311 r^2)/24 + (13877 r)/12 + 3125
n=6     r^4/24 + (131 r^3)/12 + (9527 r^2)/24 + (53785 r)/12 + 15625
n=7     r^4/24 + (259 r^3)/12 + (27791 r^2)/24 + (210245 r)/12 + 78125
n=8     r^4/24 + (515 r^3)/12 + (81815 r^2)/24 + (826825 r)/12 + 390625
n=9     r^4/24 + (1027 r^3)/12 + (242351 r^2)/24 + (3265877 r)/12 + 1953125
n=10    r^4/24 + (2051 r^3)/12 + (720887 r^2)/24 + (12941305 r)/12 + 9765625
n=11    r^4/24 + (4099 r^3)/12 + (2150351 r^2)/24 + (51402725 r)/12 + 48828125
n=12    r^4/24 + (8195 r^3)/12 + (6426455 r^2)/24 + (204531625 r)/12 + 244140625
n=13    r^4/24 + (16387 r^3)/12 + (19230191 r^2)/24 + (814905077 r)/12 + 1220703125
n=14    r^4/24 + (32771 r^3)/12 + (57592247 r^2)/24 + (3249988825 r)/12 + 6103515625

 

x=6

n=1     r^5/120 + r^4/6 + (31 r^3)/24 + (29 r^2)/6 + (87 r)/10 + 6     A000389  6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
n=2     r^5/120 + r^4/4 + (67 r^3)/24 + (59 r^2)/4 + (186 r)/5 + 36     A005583  Coefficients of Chebyshev polynomials
n=3     r^5/120 + (5 r^4)/12 + (163 r^3)/24 + (595 r^2)/12 + (841 r)/5 + 216
n=4     r^5/120 + (3 r^4)/4 + (427 r^3)/24 + (705 r^2)/4 + (3921 r)/5 + 1296
n=5     r^5/120 + (17 r^4)/12 + (1171 r^3)/24 + (7783 r^2)/12 + (18631 r)/5 + 7776
n=6     r^5/120 + (11 r^4)/4 + (3307 r^3)/24 + (9769 r^2)/4 + (89661 r)/5 + 46656
n=7     r^5/120 + (65 r^4)/12 + (9523 r^3)/24 + (112135 r^2)/12 + (435391 r)/5 + 279936
n=8     r^5/120 + (43 r^4)/4 + (27787 r^3)/24 + (144665 r^2)/4 + (2128221 r)/5 + 1679616
n=9     r^5/120 + (257 r^4)/12 + (81811 r^3)/24 + (1693783 r^2)/12 + (10454431 r)/5 + 10077696
n=10    r^5/120 + (171 r^4)/4 + (242347 r^3)/24 + (2217129 r^2)/4 + (51549261 r)/5 + 60466176
n=11    r^5/120 + (1025 r^4)/12 + (720883 r^3)/24 + (26239975 r^2)/12 + (254924191 r)/5 + 362797056
n=12    r^5/120 + (683 r^4)/4 + (2150347 r^3)/24 + (34624825 r^2)/4 + (1263537021 r)/5 + 2176782336
n=13    r^5/120 + (4097 r^4)/12 + (6426451 r^3)/24 + (412264183 r^2)/12 + (6273955231 r)/5 + 13060694016
n=14    r^5/120 + (2731 r^4)/4 + (19230187 r^3)/24 + (546466889 r^2)/4 + (31196658861 r)/5 + 78364164096

 

x=7

n=1    r^6/720 + (3 r^5)/80 + (59 r^4)/144 + (37 r^3)/16 + (319 r^2)/45 + (223 r)/20 + 7     A000579  7th Pascal Triangle Figurate numbers or binomial coefficients C(n,6).
n=2    r^6/720 + (13 r^5)/240 + (119 r^4)/144 + (307 r^3)/48 + (4801 r^2)/180 + (1141 r)/20 + 49    A005584  Coefficients of Chebyshev polynomials.
n=3    r^6/720 + (7 r^5)/80 + (275 r^4)/144 + (321 r^3)/16 + (9953 r^2)/90 + (6167 r)/20 + 343   
n=4    r^6/720 + (37 r^5)/240 + (695 r^4)/144 + (3259 r^3)/48 + (87241 r^2)/180 + (34349 r)/20 + 2401
n=5    r^6/720 + (23 r^5)/80 + (1859 r^4)/144 + (3857 r^3)/16 + (197963 r^2)/90 + (194943 r)/20 + 16807
n=6    r^6/720 + (133 r^5)/240 + (5159 r^4)/144 + (42427 r^3)/48 + (1839901 r^2)/180 + (1120581 r)/20 + 117649
n=7    r^6/720 + (87 r^5)/80 + (14675 r^4)/144 + (53041 r^3)/16 + (4347953 r^2)/90 + (6500647 r)/20 + 823543
n=8    r^6/720 + (517 r^5)/240 + (42455 r^4)/144 + (606619 r^3)/48 + (41615041 r^2)/180 + (37969309 r)/20 + 5764801
n=9    r^6/720 + (343 r^5)/80 + (124259 r^4)/144 + (780177 r^3)/16 + (100518263 r^2)/90 + (222933263 r)/20 + 40353607
n=10   r^6/720 + (2053 r^5)/240 + (366599 r^4)/144 + (9111547 r^3)/48 + (978226501 r^2)/180 + (1314269621 r)/20 + 282475249
n=11   r^6/720 + (1367 r^5)/80 + (1087475 r^4)/144 + (11902961 r^3)/16 + (2393393453 r^2)/90 + (7773078327 r)/20 + 1977326743
n=12   r^6/720 + (8197 r^5)/240 + (3237815 r^4)/144 + (140652379 r^3)/48 + (23526756841 r^2)/180 + (46091574669 r)/20 + 13841287201
n=13   r^6/720 + (5463 r^5)/80 + (9664259 r^4)/144 + (185372497 r^3)/16 + (58017612563 r^2)/90 + (273876308383 r)/20 + 96889010407
n=14   r^6/720 + (32773 r^5)/240 + (28894439 r^4)/144 + (2205108667 r^3)/48 + (573871421101 r^2)/180 + (1630131988261 r)/20 + 678223072849

Expand[Table [SeriesAtLevelR, {r,-1,5},{x,x,x},{n,1,14}]]//TableForm

see also these formulas fully simplified or these formulas factored or a chart of initial numerical values

 

r=-1

n=1   1    ones or    1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).
n=2   2 x - 1    A005408  The odd numbers.
n=3   3 x^2 - 3 x + 1   A003215 Hex (or centered hexagonal) numbers (crystal ball sequence for hexagonal lattice).
n=4   4 x^3 - 6 x^2 + 4 x - 1   A005917  Rhombic dodecahedral numbers
n=5   5 x^4 - 10 x^3 + 10 x^2 - 5 x + 1    A022521  Nexus numbers for the power of 5.
n=6   6 x^5 - 15 x^4 + 20 x^3 - 15 x^2 + 6 x - 1    A022522  Nexus numbers for the power of 6.
n=7   7 x^6 - 21 x^5 + 35 x^4 - 35 x^3 + 21 x^2 - 7 x + 1    A022523  Nexus numbers for the power of 7.
n=8   8 x^7 - 28 x^6 + 56 x^5 - 70 x^4 + 56 x^3 - 28 x^2 + 8 x - 1    A022524  Nexus numbers for the power of 8.
n=9   9 x^8 - 36 x^7 + 84 x^6 - 126 x^5 + 126 x^4 - 84 x^3 + 36 x^2 - 9 x + 1    A022525  Nexus numbers for the power of 9.
n=10   10 x^9 - 45 x^8 + 120 x^7 - 210 x^6 + 252 x^5 - 210 x^4 + 120 x^3 - 45 x^2 + 10 x - 1    A022526  Nexus numbers for the power of 10.
n=11   11 x^10 - 55 x^9 + 165 x^8 - 330 x^7 + 462 x^6 - 462 x^5 + 330 x^4 - 165 x^3 + 55 x^2 - 11 x + 1   A022527  Nexus numbers for the power of 11.
n=12   12 x^11 - 66 x^10 + 220 x^9 - 495 x^8 + 792 x^7 - 924 x^6 + 792 x^5 - 495 x^4 + 220 x^3 - 66 x^2 + 12 x - 1   A022528  Nexus numbers for the power of 12.
n=13   13 x^12 - 78 x^11 + 286 x^10 - 715 x^9 + 1287 x^8 - 1716 x^7 + 1716 x^6 - 1287 x^5 + 715 x^4 - 286 x^3 + 78 x^2 - 13 x + 1   A022529  Nexus numbers for the power of 13.
n=14   14 x^13 - 91 x^12 + 364 x^11 - 1001 x^10 + 2002 x^9 - 3003 x^8 + 3432 x^7 - 3003 x^6 + 2002 x^5 - 1001 x^4 + 364 x^3 - 91 x^2 + 14 x - 1   A022530  Nexus numbers for the power of 14.
 

r=0

n=1   x    Integers or the power of 1.
n=2   x^2   A000290  The squares.
n=3   x^3   A000578  The cubes.
n=4   x^4   A000583  The fourth power.  "Figurate numbers based on 4-dimensional regular convex polytope
              called the 4-measure polytope, 4-hypercube or tessaract with
              Schlafli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004"
n=5   x^5   A000584  The fifth power.
n=6   x^6   A001014  The sixth power.  Numbers both square and cubic - pdg(AT)worldofnumbers.com.
n=7   x^7   A001015  The seventh power.
n=8   x^8   A001016  The eighth power.
n=9   x^9   A001017   The ninth power.
n=10   x^10  A008454  The tenth power.
n=11   x^11   A008455  The eleventh power.
n=12   x^12   A008456  The twelfth power.
n=13   x^13   A010801  The 13th power.
n=14   x^14   A010802  The 14th power.
 

r=1

n=1   x^2/2 + x/2     A000217    3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
n=2   x^3/3 + x^2/2 + x/6   A000330  Square pyramidal numbers. 
n=3   x^4/4 + x^3/2 + x^2/4   A000537  Sum of first n cubes; or n-th triangular number squared.
n=4   x^5/5 + x^4/2 + x^3/3 - x/30   A000538  Sum of fourth powers: 0^4+1^4+...+n^4.
n=5   x^6/6 + x^5/2 + (5 x^4)/12 - x^2/12   A000539  Sum of 5th powers: 1^5 + 2^5 + ... + n^5.
n=6   x^7/7 + x^6/2 + x^5/2 - x^3/6 + x/42   A000540  Sum of 6th powers: 1^6 + 2^6 + ... + n^6.
n=7   x^8/8 + x^7/2 + (7 x^6)/12 - (7 x^4)/24 + x^2/12  A000541  Sum of 7th powers: 1^7 + 2^7 + ... + n^7.
n=8   x^9/9 + x^8/2 + (2 x^7)/3 - (7 x^5)/15 + (2 x^3)/9 - x/30  A000542  Sum of 8th powers: 1^8 + 2^8 + ... + n^8.
n=9   x^10/10 + x^9/2 + (3 x^8)/4 - (7 x^6)/10 + x^4/2 - (3 x^2)/20   A007487  Sum of 9th powers.
n=10   x^11/11 + x^10/2 + (5 x^9)/6 - x^7 + x^5 - x^3/2 + (5 x)/66   A023002  Sum of 10th powers.
n=11   x^12/12 + x^11/2 + (11 x^10)/12 - (11 x^8)/8 + (11 x^6)/6 - (11 x^4)/8 + (5 x^2)/12   Sum of 11th powers.
n=12   x^13/13 + x^12/2 + x^11 - (11 x^9)/6 + (22 x^7)/7 - (33 x^5)/10 + (5 x^3)/3 - (691 x)/2730   Sum of 12th powers.
n=13   x^14/14 + x^13/2 + (13 x^12)/12 - (143 x^10)/60 + (143 x^8)/28 - (143 x^6)/20 + (65 x^4)/12 - (691 x^2)/420  Sum of 13th powers.
n=14   x^15/15 + x^14/2 + (7 x^13)/6 - (91 x^11)/30 + (143 x^9)/18 - (143 x^7)/10 + (91 x^5)/6 - (691 x^3)/90 + (7 x)/6   Sum of 14th powers.
 

r=2

n=1   x^3/6 + x^2/2 + x/3  A000292   4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).

n=2   x^4/12 + x^3/3 + (5 x^2)/12 + x/6       A002415  4-dimensional pyramidal numbers: n^2*(n^2-1)/12.

                                     Also number of ways to legally insert two pairs of parentheses into a
                                            string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827)
                                            ways to insert the parentheses, but we must subtract 2(m+1) for illegal
                                            clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2)
                                            for 2 clumps of 2 parentheses, and (m-1)C(m+1,2) for 1 clump of 2
                                            parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.)  E.g. for n=2

                                     there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)),a((b)).
n=3   x^5/20 + x^4/4 + (5 x^3)/12 + x^2/4 + x/30   A085437  or  A024166  Sum of (j-i)^3 for 1 <= i < j <= n.
n=4   x^6/30 + x^5/5 + (5 x^4)/12 + x^3/3 + x^2/20 - x/30  
n=5   x^7/42 + x^6/6 + (5 x^5)/12 + (5 x^4)/12 + x^3/12 - x^2/12 - x/42
n=6   x^8/56 + x^7/7 + (5 x^6)/12 + x^5/2 + x^4/8 - x^3/6 - (5 x^2)/84 + x/42
n=7   x^9/72 + x^8/8 + (5 x^7)/12 + (7 x^6)/12 + (7 x^5)/40 - (7 x^4)/24 - (5 x^3)/36 + x^2/12 + x/30
n=8   x^10/90 + x^9/9 + (5 x^8)/12 + (2 x^7)/3 + (7 x^6)/30 - (7 x^5)/15 - (5 x^4)/18 + (2 x^3)/9 + (7 x^2)/60 - x/30
n=9   x^11/110 + x^10/10 + (5 x^9)/12 + (3 x^8)/4 + (3 x^7)/10 - (7 x^6)/10 - x^5/2 + x^4/2 + (7 x^3)/20 - (3 x^2)/20 - (5 x)/66
n=10   x^12/132 + x^11/11 + (5 x^10)/12 + (5 x^9)/6 + (3 x^8)/8 - x^7 - (5 x^6)/6 + x^5 + (7 x^4)/8 - x^3/2 - (15 x^2)/44 + (5 x)/66
n=11   x^13/156 + x^12/12 + (5 x^11)/12 + (11 x^10)/12 + (11 x^9)/24 - (11 x^8)/8 - (55 x^7)/42 + (11 x^6)/6 + (77 x^5)/40 - (11 x^4)/8 - (5 x^3)/4 + (5 x^2)/12 + (691 x)/2730
n=12   x^14/182 + x^13/13 + (5 x^12)/12 + x^11 + (11 x^10)/20 - (11 x^9)/6 - (55 x^8)/28 + (22 x^7)/7 + (77 x^6)/20 - (33 x^5)/10 - (15 x^4)/4 + (5 x^3)/3 + (7601 x^2)/5460 - (691 x)/2730
n=13   x^15/210 + x^14/14 + (5 x^13)/12 + (13 x^12)/12 + (13 x^11)/20 - (143 x^10)/60 - (715 x^9)/252 ... (143 x^7)/20 - (143 x^6)/20 - (39 x^5)/4 + (65 x^4)/12 + (7601 x^3)/1260 - (691 x^2)/420 - (7 x)/6
n=14   x^16/240 + x^15/15 + (5 x^14)/12 + (7 x^13)/6 + (91 x^12)/120 - (91 x^11)/30 - (143 x^10)/36 + ... 0 - (143 x^7)/10 - (91 x^6)/4 + (91 x^5)/6 + (7601 x^4)/360 - (691 x^3)/90 - (91 x^2)/12 + (7 x)/6
 

r=3

n=1   x^4/24 + x^3/4 + (11 x^2)/24 + x/4     A000332  5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
n=2   x^5/60 + x^4/8 + x^3/3 + (3 x^2)/8 + (3 x)/20   A005585  5-dimensional pyramidal numbers: n(n+1) ... (n+3)(2n+3)/5!.
n=3   x^6/120 + (3 x^5)/40 + x^4/4 + (3 x^3)/8 + (29 x^2)/120 + x/20  
n=4   x^7/210 + x^6/20 + x^5/5 + (3 x^4)/8 + (19 x^3)/60 + (3 x^2)/40 - (3 x)/140
n=5   x^8/336 + x^7/28 + x^6/6 + (3 x^5)/8 + (19 x^4)/48 + x^3/8 - (11 x^2)/168 - x/28
n=6   x^9/504 + (3 x^8)/112 + x^7/7 + (3 x^6)/8 + (19 x^5)/40 + (3 x^4)/16 - (8 x^3)/63 - (5 x^2)/56 + x/140
n=7   x^10/720 + x^9/48 + x^8/8 + (3 x^7)/8 + (133 x^6)/240 + (21 x^5)/80 - (2 x^4)/9 - (5 x^3)/24 + x^2/24 + x/20
n=8   x^11/990 + x^10/60 + x^9/9 + (3 x^8)/8 + (19 x^7)/30 + (7 x^6)/20 - (16 x^5)/45 - (5 x^4)/12 + (19 x^3)/180 + (7 x^2)/40 + x/220
n=9   x^12/1320 + (3 x^11)/220 + x^10/10 + (3 x^9)/8 + (57 x^8)/80 + (9 x^7)/20 - (8 x^6)/15 - (3 x^5)/4 + (19 x^4)/80 + (21 x^3)/40 - (23 x^2)/1320 - (5 x)/44
n=10   x^13/1716 + x^12/88 + x^11/11 + (3 x^10)/8 + (19 x^9)/24 + (9 x^8)/16 - (16 x^7)/21 - (5 x^6)/4 + (19 x^5)/40 + (21 x^4)/16 - x^3/22 - (45 x^2)/88 - (1017 x)/20020
n=11   x^14/2184 + x^13/104 + x^12/12 + (3 x^11)/8 + (209 x^10)/240 + (11 x^9)/16 - (22 x^8)/21 - (55 x^7)/28 + (209 x^6)/240 + (231 x^5)/80 - x^4/8 - (15 x^3)/8 - (1669 x^2)/10920 + (691 x)/1820
n=12   x^15/2730 + (3 x^14)/364 + x^13/13 + (3 x^12)/8 + (19 x^11)/20 + (33 x^10)/40 - (88 x^9)/63 - ... 7)/140 + (231 x^6)/40 - (3 x^5)/10 - (45 x^4)/8 - (2141 x^3)/3276 + (7601 x^2)/3640 + (601 x)/1820
n=13   x^16/3360 + x^15/140 + x^14/14 + (3 x^13)/8 + (247 x^12)/240 + (39 x^11)/40 - (572 x^10)/315 - ... x^7)/40 - (13 x^6)/20 - (117 x^5)/8 - (2141 x^4)/1008 + (7601 x^3)/840 + (1313 x^2)/840 - (7 x)/4
n=14   x^17/4080 + x^16/160 + x^15/15 + (3 x^14)/8 + (133 x^13)/120 + (91 x^12)/80 - (104 x^11)/45 - ... x^7)/10 - (273 x^6)/8 - (2141 x^5)/360 + (7601 x^4)/240 + (337 x^3)/45 - (91 x^2)/8 - (809 x)/340
 

r=4

n=1   x^5/120 + x^4/12 + (7 x^3)/24 + (5 x^2)/12 + x/5     A000389  6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
n=2   x^6/360 + x^5/30 + (11 x^4)/72 + x^3/3 + (31 x^2)/90 + (2 x)/15   A040977  C(n+5,5)*(n+3)/3; Sequence is n^2*(n^2-1)*(n^2-4)/360 if offset 3.
n=3   x^7/840 + x^6/60 + (11 x^5)/120 + x^4/4 + (7 x^3)/20 + (7 x^2)/30 + (2 x)/35   
n=4   x^8/1680 + x^7/105 + (11 x^6)/180 + x^5/5 + (251 x^4)/720 + (3 x^3)/10 + (113 x^2)/1260 - x/105
n=5   x^9/3024 + x^8/168 + (11 x^7)/252 + x^6/6 + (251 x^5)/720 + (3 x^4)/8 + (55 x^3)/378 - x^2/21 - (4 x)/105
n=6    x^10/5040 + x^9/252 + (11 x^8)/336 + x^7/7 + (251 x^6)/720 + (9 x^5)/20 + (221 x^4)/1008 - (11 x^3)/126 - (127 x^2)/1260 - x/105
n=7   x^11/7920 + x^10/360 + (11 x^9)/432 + x^8/8 + (251 x^7)/720 + (21 x^6)/40 + (221 x^5)/720 - (11 x^4)/72 - (31 x^3)/135 + (8 x)/165
n=8   x^12/11880 + x^11/495 + (11 x^10)/540 + x^9/9 + (251 x^8)/720 + (3 x^7)/5 + (221 x^6)/540 - (11 x^5)/45 - (199 x^4)/432 - x^3/90 + (361 x^2)/1980 + (7 x)/165
n=9   x^13/17160 + x^12/660 + x^11/60 + x^10/10 + (251 x^9)/720 + (27 x^8)/40 + (221 x^7)/420 - (11 x^6)/30 - (199 x^5)/240 - x^4/40 + (529 x^3)/990 + (19 x^2)/165 - (44 x)/455
n=10   x^14/24024 + x^13/858 + x^12/72 + x^11/11 + (251 x^10)/720 + (3 x^9)/4 + (221 x^8)/336 - (11 x^7)/21 - (199 x^6)/144 - x^5/20 + (707 x^4)/528 + (9 x^3)/22 - (1117 x^2)/2340 - (2663 x)/15015
n=11   x^15/32760 + x^14/1092 + (11 x^13)/936 + x^12/12 + (251 x^11)/720 + (33 x^10)/40 + (2431 x^9)/ ... 008 - (11 x^6)/120 + (707 x^5)/240 + (9 x^4)/8 - (20984 x^3)/12285 - (986 x^2)/1365 + (368 x)/1365
n=12   x^16/43680 + x^15/1365 + (11 x^14)/1092 + x^13/13 + (251 x^12)/720 + (9 x^11)/10 + (2431 x^10) ... 07 x^6)/120 + (27 x^5)/10 - (22429 x^4)/4368 - (4871 x^3)/1638 + (27473 x^2)/16380 + (1247 x)/1365
n=13   x^17/57120 + x^16/1680 + (11 x^15)/1260 + x^14/14 + (251 x^13)/720 + (39 x^12)/40 + (2873 x^11 ... + (117 x^6)/20 - (22429 x^5)/1680 - (4871 x^4)/504 + (13369 x^3)/1890 + (167 x^2)/35 - (244 x)/255
n=14   x^18/73440 + x^17/2040 + (11 x^16)/1440 + x^15/15 + (251 x^14)/720 + (21 x^13)/20 + (2873 x^12 ... 429 x^6)/720 - (4871 x^5)/180 + (53581 x^4)/2160 + (2039 x^3)/90 - (22649 x^2)/3060 - (1511 x)/255
 

r=5

n=1   x^6/720 + x^5/48 + (17 x^4)/144 + (5 x^3)/16 + (137 x^2)/360 + x/6      A000579  7th Pascal Triangle Figurate numbers or binomial coefficients C(n,6).
n=2   x^7/2520 + x^6/144 + (7 x^5)/144 + (25 x^4)/144 + (239 x^3)/720 + (23 x^2)/72 + (5 x)/42   A050486  C(n+6,6)*(2n+7)/7.
n=3   x^8/6720 + x^7/336 + (7 x^6)/288 + (5 x^5)/48 + (721 x^4)/2880 + x^3/3 + (227 x^2)/1008 + (5 x)/84  
n=4   x^9/15120 + x^8/672 + x^7/72 + (5 x^6)/72 + x^5/5 + (95 x^4)/288 + (865 x^3)/3024 + (25 x^2)/252
n=5   x^10/30240 + (5 x^9)/6048 + (5 x^8)/576 + (25 x^7)/504 + x^6/6 + (95 x^5)/288 + (4313 x^4)/12096 + (235 x^3)/1512 - (23 x^2)/720 - x/28
n=6   x^11/55440 + x^10/2016 + (5 x^9)/864 + (25 x^8)/672 + x^7/7 + (95 x^6)/288 + (863 x^5)/2016 + (475 x^4)/2016 - (119 x^3)/2160 - (13 x^2)/126 - (5 x)/231
n=7   x^12/95040 + x^11/3168 + (7 x^10)/1728 + (25 x^9)/864 + x^8/8 + (95 x^7)/288 + (863 x^6)/1728 + (95 x^5)/288 - (821 x^4)/8640 - (49 x^3)/216 - (53 x^2)/1584 + (5 x)/132
n=8   x^13/154440 + x^12/4752 + (7 x^11)/2376 + (5 x^10)/216 + x^9/9 + (95 x^8)/288 + (863 x^7)/1512 + (95 x^6)/216 - (329 x^5)/2160 - (395 x^4)/864 - (157 x^3)/1584 + (65 x^2)/396 + (200 x)/3003
n=9   x^14/240240 + x^13/6864 + (7 x^12)/3168 + (5 x^11)/264 + x^10/10 + (95 x^9)/288 + (863 x^8)/13 ... 29 x^6)/1440 - (79 x^5)/96 - (1433 x^4)/6336 + (365 x^3)/792 + (151577 x^2)/720720 - (629 x)/12012
n=10   x^15/360360 + (5 x^14)/48048 + (35 x^13)/20592 + (25 x^12)/1584 + x^11/11 + (95 x^10)/288 + (4 ... 5 x^6)/288 - (159 x^5)/352 + (1225 x^4)/1056 + (1549789 x^3)/2162160 - (439 x^2)/1287 - (35 x)/143
n=11   x^16/524160 + x^15/13104 + (5 x^14)/3744 + (25 x^13)/1872 + x^12/12 + (95 x^11)/288 + (9493 x^ ... 53 x^6)/64 + (245 x^5)/96 + (222583 x^4)/112320 - (22511 x^3)/19656 - (2011 x^2)/1872 + (15 x)/364
n=12   x^17/742560 + x^16/17472 + x^15/936 + (25 x^14)/2184 + x^13/13 + (95 x^12)/288 + (863 x^11)/10 ... ^6)/48 + (51913 x^5)/10920 - (30245 x^4)/8736 - (24277 x^3)/5616 + (2335 x^2)/3276 + (5350 x)/4641
n=13   x^18/1028160 + x^17/22848 + x^16/1152 + (5 x^15)/504 + x^14/14 + (95 x^13)/288 + (11219 x^12)/ ... x^6)/5040 - (6049 x^5)/672 - (24361 x^4)/1728 + (3935 x^3)/1512 + (546863 x^2)/85680 + (107 x)/204
n=14   x^19/1395360 + x^18/29376 + (7 x^17)/9792 + (5 x^16)/576 + x^15/15 + (95 x^14)/288 + (863 x^13 ... x^6)/288 - (34097 x^5)/864 + (7975 x^4)/864 + (364283 x^3)/12240 + (139 x^2)/306 - (46705 x)/6783

Factor[Table [SeriesAtLevelR, {r,-1,5},{x,x,x},{n,1,14}]]//TableForm

see also these formulas fully simplified or these formulas expanded or a chart of initial numerical values

 

r=-1

n=1    1
n=2    2 x - 1
n=3    3 x^2 - 3 x + 1
n=4    (2 x - 1) (2 x^2 - 2 x + 1)
n=5     5 x^4 - 10 x^3 + 10 x^2 - 5 x + 1
n=6    (2 x - 1) (x^2 - x + 1) (3 x^2 - 3 x + 1)
n=7    7 x^6 - 21 x^5 + 35 x^4 - 35 x^3 + 21 x^2 - 7 x + 1
n=8    (2 x - 1) (2 x^2 - 2 x + 1) (2 x^4 - 4 x^3 + 6 x^2 - 4 x + 1)
n=9    (3 x^2 - 3 x + 1) (3 x^6 - 9 x^5 + 18 x^4 - 21 x^3 + 15 x^2 - 6 x + 1)
n=10   (2 x - 1) (x^4 - 2 x^3 + 4 x^2 - 3 x + 1) (5 x^4 - 10 x^3 + 10 x^2 - 5 x + 1)
n=11   11 x^10 - 55 x^9 + 165 x^8 - 330 x^7 + 462 x^6 - 462 x^5 + 330 x^4 - 165 x^3 + 55 x^2 - 11 x + 1
n=12   (2 x - 1) (x^2 - x + 1) (2 x^2 - 2 x + 1) (3 x^2 - 3 x + 1) (x^4 - 2 x^3 + 5 x^2 - 4 x + 1)
n=13   13 x^12 - 78 x^11 + 286 x^10 - 715 x^9 + 1287 x^8 - 1716 x^7 + 1716 x^6 - 1287 x^5 + 715 x^4 - 286 x^3 + 78 x^2 - 13 x + 1
n=14   (2 x - 1) (x^6 - 3 x^5 + 9 x^4 - 13 x^3 + 11 x^2 - 5 x + 1) (7 x^6 - 21 x^5 + 35 x^4 - 35 x^3 + 21 x^2 - 7 x + 1)
 

r=0

n=1    x
n=2    x^2
n=3    x^3
n=4    x^4
n=5    x^5
n=6    x^6
n=7    x^7
n=8    x^8
n=9    x^9
n=10   x^10
n=11   x^11
n=12   x^12
n=13   x^13
n=14   x^14
 

r=1

n=1    1/2 x (x + 1)
n=2    1/6 x (x + 1) (2 x + 1)
n=3    1/4 x^2 (x + 1)^2
n=4    1/30 x (x + 1) (2 x + 1) (3 x^2 + 3 x - 1)
n=5    1/12 x^2 (x + 1)^2 (2 x^2 + 2 x - 1)
n=6    1/42 x (x + 1) (2 x + 1) (3 x^4 + 6 x^3 - 3 x + 1)
n=7    1/24 x^2 (x + 1)^2 (3 x^4 + 6 x^3 - x^2 - 4 x + 2)
n=8    1/90 x (x + 1) (2 x + 1) (5 x^6 + 15 x^5 + 5 x^4 - 15 x^3 - x^2 + 9 x - 3)
n=9    1/20 x^2 (x + 1)^2 (x^2 + x - 1) (2 x^4 + 4 x^3 - x^2 - 3 x + 3)
n=10   1/66 x (x + 1) (2 x + 1) (x^2 + x - 1) (3 x^6 + 9 x^5 + 2 x^4 - 11 x^3 + 3 x^2 + 10 x - 5)
n=11   1/24 x^2 (x + 1)^2 (2 x^8 + 8 x^7 + 4 x^6 - 16 x^5 - 5 x^4 + 26 x^3 - 3 x^2 - 20 x + 10)
n=12   (x (x + 1) (2 x + 1) (105 x^10 + 525 x^9 + 525 x^8 - 1050 x^7 - 1190 x^6 + 2310 x^5 + 1420 x^4 - 3285 x^3 - 287 x^2 + 2073 x - 691))/2730
n=13   1/420 x^2 (x + 1)^2 (30 x^10 + 150 x^9 + 125 x^8 - 400 x^7 - 326 x^6 + 1052 x^5 + 367 x^4 - 1786 x^3 + 202 x^2 + 1382 x - 691)
n=14   1/90 x (x + 1) (2 x + 1) (3 x^12 + 18 x^11 + 24 x^10 - 45 x^9 - 81 x^8 + 144 x^7 + 182 x^6 - 345 x^5 - 217 x^4 + 498 x^3 + 44 x^2 - 315 x + 105)
 

r=2

n=1    1/6 x (x + 1) (x + 2)
n=2    1/12 x (x + 1)^2 (x + 2)
n=3    1/60 x (x + 1) (x + 2) (3 x^2 + 6 x + 1)
n=4    1/60 x (x + 1)^2 (x + 2) (2 x^2 + 4 x - 1)
n=5    1/84 x (x + 1) (x + 2) (x^2 + 2 x - 1) (2 x^2 + 4 x + 1)
n=6    1/168 x (x + 1)^2 (x + 2) (x^2 + 2 x - 1) (3 x^2 + 6 x - 2)
n=7    1/360 x (x + 1) (x + 2) (5 x^6 + 30 x^5 + 50 x^4 - 37 x^2 + 6 x + 6)
n=8    1/180 x (x + 1)^2 (x + 2) (2 x^2 + 4 x - 1) (x^4 + 4 x^3 + x^2 - 6 x + 3)
n=9    1/660 x (x + 1) (x + 2) (x^2 + x - 1) (x^2 + 3 x + 1) (6 x^4 + 24 x^3 + 5 x^2 - 38 x + 25)
n=10   1/264 x (x + 1)^2 (x + 2) (x^2 + 2 x - 2) (2 x^6 + 12 x^5 + 16 x^4 - 16 x^3 - 17 x^2 + 30 x - 5)
n=11   (x (x + 1) (x + 2) (70 x^10 + 700 x^9 + 2310 x^8 + 1680 x^7 - 4655 x^6 - 4410 x^5 + 8240 x^4 + 4120 x^3 - 7819 x^2 + 202 x + 1382))/10920
n=12   (x (x + 1)^2 (x + 2) (30 x^10 + 300 x^9 + 925 x^8 + 200 x^7 - 3022 x^6 - 772 x^5 + 7073 x^4 - 1228 x^3 - 7888 x^2 + 5528 x - 691))/5460
n=13   (x (x + 1) (x + 2) (6 x^12 + 72 x^11 + 297 x^10 + 330 x^9 - 765 x^8 - 1368 x^7 + 2059 x^6 + 2994 x^5 - 4091 x^4 - 2724 x^3 + 4069 x^2 + 66 x - 735))/1260
n=14   1/720 x (x + 1)^2 (x + 2) (3 x^12 + 36 x^11 + 141 x^10 + 90 x^9 - 591 x^8 - 552 x^7 + 2123 x^6 + 1170 x^5 - 5182 x^4 + 336 x^3 + 5846 x^2 - 3780 x + 420)
 

r=3

n=1    1/24 x (x + 1) (x + 2) (x + 3)
n=2    1/120 x (x + 1) (x + 2) (x + 3) (2 x + 3)
n=3    1/120 x (x + 1) (x + 2) (x + 3) (x^2 + 3 x + 1)
n=4    1/840 x (x + 1) (x + 2) (x + 3) (2 x + 3) (2 x^2 + 6 x - 1)
n=5    1/336 x (x + 1) (x + 2) (x + 3) (x^2 + 2 x - 1) (x^2 + 4 x + 2)
n=6    (x (x + 1) (x + 2) (x + 3) (2 x + 3) (5 x^4 + 30 x^3 + 35 x^2 - 30 x + 2))/5040
n=7    1/720 x (x + 1) (x + 2) (x + 3) (x^6 + 9 x^5 + 25 x^4 + 15 x^3 - 20 x^2 - 6 x + 6)
n=8    (x (x + 1) (x + 2) (x + 3) (2 x + 3) (2 x^6 + 18 x^5 + 45 x^4 - 69 x^2 + 36 x + 1))/3960
n=9    (x (x + 1) (x + 2) (x + 3) (2 x^8 + 24 x^7 + 98 x^6 + 126 x^5 - 97 x^4 - 204 x^3 + 127 x^2 + 84 x - 50))/2640
n=10   (x (x + 1) (x + 2) (x + 3) (2 x + 3) (70 x^8 + 840 x^7 + 3220 x^6 + 2520 x^5 - 7735 x^4 - 4830 x^3 + 13725 x^2 - 5130 x - 678))/240240
n=11   (x (x + 1) (x + 2) (x + 3) (10 x^10 + 150 x^9 + 810 x^8 + 1620 x^7 - 511 x^6 - 4599 x^5 + 615 x^4 + 7065 x^3 - 2542 x^2 - 3090 x + 1382))/21840
n=12   (x (x + 1) (x + 2) (x + 3) (2 x + 3) (6 x^10 + 90 x^9 + 465 x^8 + 720 x^7 - 1218 x^6 - 3024 x^5 + 3775 x^4 + 4830 x^3 - 8270 x^2 + 2298 x + 601))/32760
n=13   (x (x + 1) (x + 2) (x + 3) (3 x^12 + 54 x^11 + 363 x^10 + 990 x^9 + 117 x^8 - 3942 x^7 - 1879 x^6 + 11034 x^5 + 2570 x^4 - 17412 x^3 + 3446 x^2 + 8016 x - 2940))/10080
n=14   (x (x + 1) (x + 2) (x + 3) (2 x + 3) (3 x^12 + 54 x^11 + 351 x^10 + 810 x^9 - 771 x^8 - 4878 x^7 + 2333 x^6 + 17190 x^5 - 12702 x^4 - 27396 x^3 + 35826 x^2 - 7380 x - 3236))/24480
 

r=4

n=1    1/120 x (x + 1) (x + 2) (x + 3) (x + 4)
n=2    1/360 x (x + 1) (x + 2)^2 (x + 3) (x + 4)
n=3    1/840 x (x + 1) (x + 2) (x + 3) (x + 4) (x^2 + 4 x + 2)
n=4    (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (3 x^2 + 12 x - 1))/5040
n=5    (x (x + 1) (x + 2) (x + 3) (x + 4) (5 x^4 + 40 x^3 + 85 x^2 + 20 x - 24))/15120
n=6    (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x^4 + 8 x^3 + 14 x^2 - 8 x - 1))/5040
n=7    (x (x + 1) (x + 2) (x + 3) (x + 4) (3 x^6 + 36 x^5 + 140 x^4 + 160 x^3 - 89 x^2 - 100 x + 48))/23760
n=8    (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x^2 + 4 x + 1) (2 x^4 + 16 x^3 + 20 x^2 - 48 x + 21))/23760
n=9    (x (x + 1) (x + 2) (x + 3) (x + 4) (42 x^8 + 672 x^7 + 3822 x^6 + 8232 x^5 + 553 x^4 - 14392 x^3 + 473 x^2 + 9508 x - 2904))/720720
n=10   (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (30 x^8 + 480 x^7 + 2600 x^6 + 4320 x^5 - 4909 x^4 - 11112 x^3 + 13544 x^2 - 288 x - 2663))/720720
n=11   (x (x + 1) (x + 2) (x + 3) (x + 4) (6 x^10 + 120 x^9 + 900 x^8 + 2880 x^7 + 2079 x^6 - 7308 x^5 - 7270 x^4 + 13840 x^3 + 4699 x^2 - 10516 x + 2208))/196560
n=12   (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (3 x^10 + 60 x^9 + 435 x^8 + 1200 x^7 - 215 x^6 - 5268 x^5 + 633 x^4 + 12424 x^3 - 9538 x^2 - 1864 x + 2494))/131040
n=13   (x (x + 1) (x + 2) (x + 3) (x + 4) (9 x^12 + 216 x^11 + 2013 x^10 + 8580 x^9 + 11943 x^8 - 24336 x^7 - 65865 x^6 + 65556 x^5 + 161038 x^4 - 158800 x^3 - 120478 x^2 + 144904 x - 20496))/514080
n=14   (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x^12 + 24 x^11 + 218 x^10 + 840 x^9 + 528 x^8 - 4608 x^7 - 5046 x^6 + 19320 x^5 + 9089 x^4 - 49272 x^3 + 26080 x^2 + 12096 x - 9066))/73440
 

r=5

n=1    1/720 x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5)
n=2    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5))/5040
n=3    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (3 x^2 + 15 x + 10))/20160
n=4    (x^2 (x + 1) (x + 2) (x + 3) (x + 4) (x + 5)^2 (2 x + 5))/30240
n=5    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x^2 + 5 x - 2) (2 x^2 + 10 x + 9))/60480
n=6    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (x^2 + 5 x - 3) (3 x^2 + 15 x + 4))/332640
n=7    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x^2 + 5 x - 3) (x^2 + 5 x - 2) (x^2 + 5 x + 5))/95040
n=8    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (14 x^6 + 210 x^5 + 980 x^4 + 1050 x^3 - 1771 x^2 - 105 x + 480))/4324320
n=9    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (12 x^8 + 240 x^7 + 1750 x^6 + 5250 x^5 + 3500 x^4 - 8750 x^3 - 4665 x^2 + 7925 x - 1258))/2882880
n=10   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (6 x^8 + 120 x^7 + 840 x^6 + 2100 x^5 - 609 x^4 - 6090 x^3 + 3980 x^2 + 2275 x - 1764))/4324320
n=11   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (3 x^10 + 75 x^9 + 720 x^8 + 3150 x^7 + 4893 x^6 - 5355 x^5 - 16520 x^4 + 11050 x^3 + 18938 x^2 - 15310 x + 540))/1572480
n=12   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (9 x^10 + 225 x^9 + 2100 x^8 + 8250 x^7 + 6489 x^6 - 33915 x^5 - 29550 x^4 + 102000 x^3 - 25606 x^2 - 53030 x + 25680))/13366080
n=13   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x^12 + 60 x^11 + 715 x^10 + 4125 x^9 + 10182 x^8 ... 2610 x^7 - 50325 x^6 - 10875 x^5 + 143534 x^4 - 14660 x^3 - 175130 x^2 + 88850 x + 8988))/2056320
n=14   (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (7 x^12 + 210 x^11 + 2450 x^10 + 13125 x^ ... x^7 - 181860 x^6 + 225225 x^5 + 571928 x^4 - 930720 x^3 - 59780 x^2 + 616350 x - 224184))/19535040

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