FLINT3.7.0-dev
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fmpz_poly_eulerian_polynomial

Synopsis#include <flint/fmpz_poly.h>
void fmpz_poly_eulerian_polynomial(fmpz_poly_t poly, ulong n)source

Sets poly to the Eulerian polynomial An(x)=∑m=0nA(n,m)xmA_n(x) = \sum_{m=0}^n A(n, m) x^m, with A0=1A_0 = 1, computed by a recurrence. The coefficients are the Eulerian numbers

A(n,m)=∑k=0m+1(−1)k(n+1k)(m+1−k)n,A(n, m) = \sum_{k=0}^{m+1} (-1)^k \binom{n+1}{k} (m + 1 - k)^n,

and the polynomials have the exponential generating function (x−1)/(x−e(x−1)t)=∑nAn(x)tn/n!(x - 1)/(x - e^{(x - 1)t}) = \sum_n A_n(x) t^n / n!. They are not the Euler polynomials.

Cite this page

The FLINT team, “fmpz_poly_eulerian_polynomial (fmpz_poly.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/fmpz_poly/fmpz_poly_eulerian_polynomial.html (generated 2026-11-02).