A348068 Coefficient of x^5 in expansion of n!* Sum_{k=0..n} binomial(x,k).

1, -9, 112, -1064, 12873, -140595, 1870385, -23551110, 351042406, -5043110072, 84074954600, -1361614072000, 25218570009424, -455365645674480, 9298765013106384, -185409487083100320, 4144212593899945056, -90492302454898284864, 2199399908894486591040

Offset: 5

Author

Seiichi Manyama, Sep 27 2021

Keywords

• sign

Crossrefs

Column k=5 of A190782.

Cf. A000482, A001233, A054651, A347987, A348063, A348064, A348065.

Programs

PARI

a(n) = n!*polcoef(sum(k=5, n, binomial(x, k)), 5);

PARI

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^5/(120*(1-x))))

Python

from sympy.abc import x
from sympy import ff, expand
def A348068(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**5) for k in range(5,n+1)) # Chai Wah Wu, Sep 27 2021

Formula

E.g.f.: (log(1 + x))^5/(120 * (1 - x)).