A348065 Coefficient of x^4 in expansion of n!* Sum_{k=0..n} binomial(x,k).

1, -5, 55, -350, 3969, -31563, 408050, -3920950, 58206676, -657328100, 11111159696, -144321864960, 2747845864464, -40364369180016, 856755330487200, -14042902728462624, 329258021171239296, -5956512800554963584, 153050034289602269952, -3028534064042216488704, 84691080748928315003904

Offset: 4

Author

Seiichi Manyama, Sep 26 2021

Keywords

• sign

Crossrefs

Column k=4 of A190782.

Cf. A000482, A054651, A028341, A347987, A348063, A348064, A348068.

Programs

PARI

a(n) = n!*polcoef(sum(k=4, n, binomial(x, k)), 4);

PARI

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^4/(24*(1-x))))

Python

from sympy.abc import x
from sympy import ff, expand
def A348065(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**4) for k in range(4,n+1)) # Chai Wah Wu, Sep 27 2021

Formula

E.g.f.: (log(1 + x))^4/(24 * (1 - x)).