Gould polynomials

In mathematics the Gould polynomials G_n(x; a,b) are polynomials introduced by H. W. Gould and named by Roman in 1984. They are given by

$\exp (x{f}^{-1}(t))=\sum _{n=0}^{\mathrm{\infty }}{G}_n(x;a,b)\frac{t^n}{n!}$

where

$f(t)=e^{at}(e^{bt}-1)$ so $f^{-1}(t)=\frac{1}{b}\sum _{k=1}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{-(b+ak)/b}{k-1})\frac{t^k}{k}$