MagicNKZ Solving for z

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

Sloane numbers for equations are on the fourth table. Example depictions of orders of series' is after that.

<<DiscreteMath`Combinatorica`

Table[MagicNKZ,{n,1,10},{k,0,12},{z,z,z}]//TableForm

Expand[Table[ MagicNKZ, {n,1,10},{k,0,12},{z,z,z}]]//TableForm

FullSimplify[Table[ MagicNKZ, {n,1,10},{k,0,12},{z,z,z}]]//TableForm

Table 4. The sequences generated by the above equations: Sloane integer series numbers

solving for z	k=0	k=1	k=2	k=3	k=4	k=5	k=6	k=7
n=1	ones 1st Pascal Triangle Figurate numbers or binomial coefficients C(n,0).	integers 2nd Pascal Triangle Figurate numbers or binomial coefficients C(n,1).	A000217 3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).	A000292 4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).	A000332 5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).	A000389 6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).	A000579 7th Pascal Triangle Figurate numbers or binomial coefficients C(n,6).	A000580 8th Pascal Triangle Figurate numbers or binomial coefficients C(n,7).
n=2	ones	integers	A000096	A005581	A005582	A005583 Coefficients of Chebyshev polynomials.	A005584 Coefficients of Chebyshev polynomials.
n=3	ones	integers	A051936 Truncated triangular numbers	A101853	A101854	A101855
n=4	ones	integers	A101859	A101860	A101861	A101862
n=5	ones	integers

example: n=1; example k=2;

Horizontal n: Read equations across rows of table. Vertical k: Read equations down columns of table.