SeriesAtLevelR Equations

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

The meaning of x The meaning of r  The meaning of n

"SeriesAtLevelR":

<<DiscreteMath`Combinatorica`

This is the "three-dimensional" version of the formula for Euler's Triangle:

for an xth value of the power level n, at r level of accumulation. 

When r = 0, a power series (in yellow) results,

when r < 0, a shell series results, and when r > 0,

a summation of powers series results. 

Click to see the numbers in matrices.

The SeriesAtLevelR Meaning of xth in series: 

(Click to see the logically reciprocal "MagicNKZ" formulas for the same series.)

solving

for x

r= -1

r=0

r=1

r=2

r=3

r=4

 r=5 r=6 r=7

n=1

ones

integers

A000217 

triangular numbers

A000292 tetrahedral numbers

A000332 5th figurate series

 A000389  6th figurate series A000579 7th figurate series  A000580 8th figurate series A000581 9th figurate series

n=2

A005408

The odds/

nexus numbers to

power of 2.

2k + 1.

A000290 

The squares.

A000330

The sums

of squares.

A002415

The sums

of sums

of squares--4D pyramidal numbers

 A005585 Sums of Sums of Sums of squares--5D pyramidal numbers A050486 C(n+6,6)* (2n+7)/7. A053347 C(n+7,7)* (n+4)/4. A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).  

n=3

A003215

Hex numbers/

nexus numbers to

power of 3.  3 k^2 + 3 k + 1

A000578 

The cubes.

A000537 

Sum of first

n cubes.

 A085437  or  A024166 Sums of Sums of cubes A101094 Sums of Sums of Sums of the 3rd power A101097 Sums of Sums of Sums of Sums of the 3rd power A101102 Sums of Sums of Sums of Sums of Sums of the 3rd power    

n=4

A005917 

Nexus numbers to

power of 4.  4 k^3 + 6 k^2 + 4 k + 1

A000583 

The fourth

power.

 A000538 Sums of the 4th power. A101089 Sums of Sums of the 4th power A101090 Sums of Sums of Sums of the 4th power A101091 Sums of Sums of Sums of Sums of the 4th power      

n=5

A022521 

Nexus numbers to

power of 5.   

 A000584 The 5th power. A000539 Sums of the 5th power. A101092 Sums of Sums of the 5th power A101099 Sums of Sums of Sums of the 5th power        
n=6 A022522 Nexus numbers to power of 6.    A001014 The 6th power. A000540 Sums of the 6th power. A101093 Sums of Sums of the 6th power          
n=7 A022523 Nexus numbers to the 7th power. A001015 The 7th power. A000541 Sums of the 7th power.            
n=8   A001016 The 8th power.              
n=9 A022525 Nexus numbers to the 9th power.                

Solving for x

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

See also the chart of initial numerical values.

If:  r  = - 1;  n = 1 through 6; then series per increasing  x are:

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.

OR 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.

OR

If:  r = 0;  n = 1 through 6; then series per increasing x are:

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.

OR

If:  r = 1;  n = 1 through 6; then series per increasing x are:

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.

OR

    OR      

If:  r = 2;  n = 1 through 6; then series per increasing x are:

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)  A101089
n=5    1/84 x (x + 1) (x + 2) (x (x + 2) - 1) (2 x (x + 2) + 1)  A101092
n=6    1/168 x (x + 1)^2 (x + 2) (x (x + 2) - 1) (3 x (x + 2) - 2) A101093
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)

OR

  

OR

 

If:  r = 3;  n = 1 through 6; then series per increasing x are:

 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) A101094
n=4    1/840 x (x + 1) (x + 2) (x + 3) (2 x + 3) (2 x (x + 3) - 1)  A101090
n=5    1/336 x (x + 1) (x + 2) (x + 3) (x (x + 2) - 1) (x (x + 4) + 2) A101099
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

OR

OR

If:  n = 1; r  = - 1 through 3; then series per increasing  x are:

n=1

r= -1    1    ones or    1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).
r=0      x      Integers or the power of 1.
r=1     1/2 x (x + 1)     A000217    3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
r=2     1/6 x (x + 1) (x + 2)  A000292   4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
r=3    1/24 x (x + 1) (x + 2) (x + 3)     A000332  5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
r=4    1/120 x (x + 1) (x + 2) (x + 3) (x + 4)     A000389  6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
r=5    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).

OR

If:  n = 2; r  = - 1 through 3; then series per increasing  x are:

n=2

r= -1   2 x - 1      A005408  The odd numbers.
r=0     x^2   A000290  The squares.
r=1    1/6 x (x + 1) (2 x + 1)   A000330  Square pyramidal numbers. 
r=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)).
r=3    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!.
r=4    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.
r=5    (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5))/5040   A050486  C(n+6,6)*(2n+7)/7.

OR

OR

If:  n = 3; r  = - 1 through 3; then series per increasing  x are:

n=3

r= -1   3 (x - 1) x + 1  A003215 Hex (or centered hexagonal) numbers (crystal ball sequence for hexagonal lattice).

r=0     x^3   A000578  The cubes.

r=1    1/4 x^2 (x + 1)^2   A000537  Sum of first n cubes; or n-th triangular number squared. 

r=2    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.

r=3    1/120 x (x + 1) (x + 2) (x + 3) (x (x + 3) + 1) A101094

r=4     1/840 x (x + 1) (x + 2) (x + 3) (x + 4) (x (x + 4) + 2) A101097

r=5     (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (3 x (x + 5) + 10))/20160

OR

OR

If:  n = 4; r  = - 1 through 3; then series per increasing  x are:

n=4

r= -1   2 x (x (2 x - 3) + 2) - 1   A005917  Rhombic dodecahedral numbers

r=0     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"

r=1    1/30 x (x + 1) (2 x + 1) (3 x (x + 1) - 1)   A000538  Sum of fourth powers: 0^4+1^4+...+n^4.

r=2    1/60 x (x + 1)^2 (x + 2) (2 x (x + 2) - 1)  A101089

r=3    1/840 x (x + 1) (x + 2) (x + 3) (2 x + 3) (2 x (x + 3) - 1)  A101090

r=4     (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (3 x (x + 4) - 1))/5040   A101091

r=5     (x^2 (x + 1) (x + 2) (x + 3) (x + 4) (x + 5)^2 (2 x + 5))/30240

OR

OR

If:  n = 5; r  = - 1 through 3; then series per increasing  x are:

n=5

r= -1   5 (x - 1) x ((x - 1) x + 1) + 1   A022521  Nexus numbers for the power of 5.

r=0     x^5  A000584  The fifth power.

r=1    1/12 x^2 (x + 1)^2 (2 x (x + 1) - 1)  A000539  Sum of 5th powers: 1^5 + 2^5 + ... + n^5.

r=2    1/84 x (x + 1) (x + 2) (x (x + 2) - 1) (2 x (x + 2) + 1) A101092

r=3    1/336 x (x + 1) (x + 2) (x + 3) (x (x + 2) - 1) (x (x + 4) + 2) A101099

r=4     (x (x + 1) (x + 2) (x + 3) (x + 4) (5 x (x + 4) (x (x + 4) + 1) - 24))/15120

r=5     (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x (x + 5) - 2) (2 x (x + 5) + 9))/60480

OR

OR

If:  n = 6; r  = - 1 through 3; then series per increasing  x are:

n=6

r= -1   (2 x - 1) ((x - 1) x + 1) (3 (x - 1) x + 1)      A022522  Nexus numbers for the power of 6.

r=0     x^6  A001014  The sixth power.  Numbers both square and cubic - pdg(AT)worldofnumbers.com.

r=1    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.

r=2    1/168 x (x + 1)^2 (x + 2) (x (x + 2) - 1) (3 x (x + 2) - 2) A101093

r=3    (x (x + 1) (x + 2) (x + 3) (2 x + 3) (5 x (x + 3) (x (x + 3) - 2) + 2))/5040

r=4     (x (x + 1) (x + 2)^2 (x + 3) (x + 4) (x (x + 4) (x (x + 4) - 2) - 1))/5040

r=5     (x (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (2 x + 5) (x (x + 5) - 3) (3 x (x + 5) + 4))/332640

OR

OR

 

The SeriesAtLevelR Meaning of rth in Series:  solving for r

 

See also the chart of initial numerical values

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

If:  x  =  1;  n = 1 through 6; then series per increasing r from r = -1 are:

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

 

If:  x  =  2;  n = 1 through 6; then series  per increasing r from r = -1 are:

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

If:  x  =  3;  n = 1 through 6; then series per increasing r from r = -1 are:

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)

OR

OR   

If:  x  =  4;  n = 1 through 6; then series per increasing r from r = -1 are:

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 r 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)

OR

  OR   

If:  x  =  5;  n = 1 through 6; then series per increasing r from r = -1 are:

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)

OR

  OR 

If:  x  =  6;  n = 1 through 6; then series per increasing r from r = -1 are:

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)

OR

 

OR

  

If:  n  =  1;  x = 1 through 6; then series per increasing r from r = -1 are:

n=1

x=1    1   ones or 1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).
x=2    r + 2     integers or the 2nd Pascal Triangle Figurate numbers or binomial coefficients C(n,1).
x=3    1/2 (r + 2) (r + 3)     A000217    3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
x=4    1/6 (r + 2) (r + 3) (r + 4)    A000292   4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
x=5    1/24 (r + 2) (r + 3) (r + 4) (r + 5)     A000332  5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
x=6    1/120 (r + 2) (r + 3) (r + 4) (r + 5) (r + 6)     A000389  6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
x=7    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).

OR

If:  n  =  2;  x = 1 through 6; then series per increasing r from r = -1 are:

n=2

x=1    1
x=2     r + 4
x=3    1/2 (r + 3) (r + 6)     A000096
x=4    1/6 (r + 3) (r + 4) (r + 8)    A005581   A class of Boolean functions of n variables and rank 2.
x=5    1/24 (r + 3) (r + 4) (r + 5) (r + 10)    A005582
x=6    1/120 (r + 3) (r + 4) (r + 5) (r + 6) (r + 12)    A005583  Coefficients of Chebyshev polynomials
x=7    1/720 (r + 3) (r + 4) (r + 5) (r + 6) (r + 7) (r + 14)    A005584  Coefficients of Chebyshev polynomials.

OR

If:  n  =  3;  x = 1 through 6; then series per increasing r from r = -1 are:

n=3

x=1    1

x=2     r + 8

x=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.

x=4    1/6 (r + 4) (r (r + 23) + 96)

x=5    1/24 (r + 4) (r + 5) (r (r + 29) + 150)

x=6    1/120 (r + 4) (r + 5) (r + 6) (r + 8) (r + 27)

x=7    1/720 (r + 4) (r + 5) (r + 6) (r + 7) (r (r + 41) + 294)

OR

OR

If:  n  =  4;  x = 1 through 6; then series per increasing r from r = -1 are:

n=4

x=1    1
x=2     r + 16
x=3    1/2 (r + 6) (r + 27)
x=4    1/6 (r + 8) (r (r + 43) + 192)
x=5    1/24 (r + 5) (r + 10) (r (r + 55) + 300)
x=6    1/120 (r + 5) (r + 6) (r + 12) (r (r + 67) + 432)
x=7    1/720 (r + 5) (r + 6) (r + 7) (r + 14) (r (r + 79) + 588)

OR

OR

If:  n  =  5;  x = 1 through 6; then series per increasing r from r = -1 are:

n=5

x=1    1
x=2     r + 32
x=3    1/2 (r (r + 65) + 486)
x=4    1/6 (r + 12) (r (r + 87) + 512)
x=5    1/24 (r (r (r (r + 134) + 3311) + 27754) + 75000)
x=6    1/120 (r + 6) (r (r (r (r + 164) + 4871) + 48604) + 155520)
x=7    1/720 (r + 6) (r + 7) (r (r (r (r + 194) + 6731) + 77914) + 288120)

OR

OR

If:  n  =  6;  x = 1 through 6; then series per increasing r from r = -1 are:

n=6

x=1    1
x=2     r + 64
x=3    1/2 (r (r + 129) + 1458)
x=4    1/6 (r + 8) (r (r + 187) + 3072)
x=5    1/24 (r + 10) (r (r (r + 252) + 7007) + 37500)
x=6    1/120 (r + 12) (r (r (r (r + 318) + 12719) + 140442) + 466560)
x=7    1/720 (r + 7) (r + 14) (r (r (r (r + 378) + 17759) + 226422) + 864360)

OR

OR

SeriesAtLevelR Meaning of nth in series:  solving for n

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 .]

If:  r = - 1; x  =  1 through 6;

then series per increasing n are:

Look down column 1 in above chart

 

If:  r = 0; x  =  1 through 6;

then series per increasing n are: 

Look down column 2 in above chart

 

If:  r = 1; x  =  1 through 6;

then series per increasing n are: 

Look down column 3 in above chart

 

If:  r = 2; x  =  1 through 6;

then series per increasing n are: 

Look down column 4 in above chart

 

If:  r = 3; x  =  1 through 6;

then series per increasing n are: 

Look down column 5 in above chart

 

If:  x  =  1;  r = - 1 through 3;

then series per increasing n are:

Look across row 1 in above chart

If:  x  =  2;  r = - 1 through 3;

then series per increasing n are: 

Look across row 2 in above chart

If:  x  =  3;  r = - 1 through 3;

then series per increasing n are: 

Look across row 3 in above chart

If:  x  =  4;  r = - 1 through 3;

then series per increasing n are: 

Look across row 4 in above chart

If:  x  =  5;  r = - 1 through 3;

then series per increasing n are: 

Look across row 5 in above chart

If:  x  =  6;  r = - 1 through 3;

then series per increasing n are: 

Look across row 6 in above chart

Send mail to Cecilia@noticingnumbers.net with questions or comments about this web site.
Copyright © 2004 noticingnumbers.net
Last modified: 12/16/05