What comes next?

[Thomas Klemm]

I'm not sure what your program does, but if you want to transform the polynomial based on binomial coefficients \(\binom{n}{k}\) to the common form based on \(n^k\) the linear transform A131689 can be used.

Example

\(f(n)=\binom{n}{1}+2\binom{n}{2}+\binom{n}{3}\)

[ 0 1 2 1 ]

[[ 1 0 0 0 ]
 [ 0 1 1 1 ]
 [ 0 0 2 6 ]
 [ 0 0 0 6 ]]

/

[ 0 .333333333333 .5 .166666666667 ]

Thus:

\(f(n)=\frac{n}{3}+\frac{n^2}{2}+\frac{n^3}{6}\)