Mahler polynomial

In mathematics, the Mahler polynomials gn(x) are polynomials introduced by in his work on the zeros of the incomplete gamma function.

Mahler polynomials are given by the generating function


 * $$\displaystyle \sum g_n(x)t^n/n! = \exp(x(1+t-e^t)) $$

Which is close to the generating function of the Touchard polynomials.

The first few examples are
 * $$g_0=1;$$
 * $$g_1=0;$$
 * $$g_2=-x;$$
 * $$g_3=-x;$$
 * $$g_4=-x+3x^2;$$
 * $$g_5=-x+10x^2;$$
 * $$g_6=-x+25x^2-15x^3;$$
 * $$g_7=-x+56x^2-105x^3;$$
 * $$g_8=-x+119x^2-490x^3+105x^4;$$