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

Sloane numbers for series generated by equations are on the second table.  

Example depictions of the orders of series' is after that.

<<DiscreteMath`Combinatorica`

SeriesAtLevelR =    

Table[SeriesAtLevelR,{n,1,6},{r,-1,2}]//TableForm

1

2 x-1

1/2 (x - 2) (x - 1) + 2 x (x - 1) + 1/2 x (x + 1)

1/6 (x - 3) (x - 2) (x - 1) + 11/6 (x - 2) x (x - 1) + 11/6 x (x + 1) (x - 1) + 1/6 x (x + 1) (x + 2)

x

1/2 (x - 1) x + 1/2 (x + 1) x

1/6 (x - 2) (x - 1) x + 2/3 (x - 1) (x + 1) x + 1/6 (x + 1) (x + 2) x

1/24 (x - 3) (x - 2) (x - 1) x + 11/24 (x - 2) (x - 1) (x + 1) x + 11/24 (x - 1) (x + 1) (x + 2) x + 1/24 (x + 1) (x + 2) (x + 3) x

1/2 x (x + 1)

1/6 (x - 1) x (x + 1) + 1/6 x (x + 2) (x + 1)

1/24 (x - 2) (x - 1) x (x + 1) + 1/6 (x - 1) x (x + 2) (x + 1) + 1/24 x (x + 2) (x + 3) (x + 1)

1/120 (x - 3) (x - 2) (x - 1) x (x + 1) + 11/120 (x - 2) (x - 1) x (x + 2) (x + 1) + 11/120 (x - 1) x (x + 2) (x + 3) (x + 1) + 1/120 x (x + 2) (x + 3) (x + 4) (x + 1)

1/6 x (x + 1) (x + 2)

1/24 (x - 1) x (x + 1) (x + 2) + 1/24 x (x + 1) (x + 3) (x + 2)

1/120 (x - 2) (x - 1) x (x + 1) (x + 2) + 1/30 (x - 1) x (x + 1) (x + 3) (x + 2) + 1/120 x (x + 1) (x + 3) (x + 4) (x + 2)

1/720 (x - 3) (x - 2) (x - 1) x (x + 1) (x + 2) + 11/720 (x - 2) (x - 1) x (x + 1) (x + 3) ( ... /720 (x - 1) x (x + 1) (x + 3) (x + 4) (x + 2) + 1/720 x (x + 1) (x + 3) (x + 4) (x + 5) (x + 2)

1/24 x (x + 1) (x + 2) (x + 3)

1/120 (x - 1) x (x + 1) (x + 2) (x + 3) + 1/120 x (x + 1) (x + 2) (x + 4) (x + 3)

1/720 (x - 2) (x - 1) x (x + 1) (x + 2) (x + 3) + 1/180 (x - 1) x (x + 1) (x + 2) (x + 4) (x + 3) + 1/720 x (x + 1) (x + 2) (x + 4) (x + 5) (x + 3)

((x - 3) (x - 2) (x - 1) x (x + 1) (x + 2) (x + 3))/5040 + (11 (x - 2) (x - 1) x (x + 1) (x ... (x + 2) (x + 4) (x + 5) (x + 3))/5040 + (x (x + 1) (x + 2) (x + 4) (x + 5) (x + 6) (x + 3))/5040

1/120 x (x + 1) (x + 2) (x + 3) (x + 4)

1/720 (x - 1) x (x + 1) (x + 2) (x + 3) (x + 4) + 1/720 x (x + 1) (x + 2) (x + 3) (x + 5) (x + 4)

((x - 2) (x - 1) x (x + 1) (x + 2) (x + 3) (x + 4))/5040 + ((x - 1) x (x + 1) (x + 2) (x + 3) (x + 5) (x + 4))/1260 + (x (x + 1) (x + 2) (x + 3) (x + 5) (x + 6) (x + 4))/5040

((x - 3) (x - 2) (x - 1) x (x + 1) (x + 2) (x + 3) (x + 4))/40320 + (11 (x - 2) (x - 1) x (x ... + 5) (x + 6) (x + 4))/40320 + (x (x + 1) (x + 2) (x + 3) (x + 5) (x + 6) (x + 7) (x + 4))/40320

Simplify[Expand[Table[SeriesAtLevelR,{n,1,6},{r,-1,3}]]]//TableForm

 

Table 3.  Sequences generated by the above equations solving for x:  Sloane numbers for series

Click to compare MagicNKZ solving for k

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.                

             

example:  r = -1;  n = 1 to 6                  example:  n = 2; r = -1 to 3

Read equations down above columns of the table//Read equations across above rows of the table

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