A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

1, -2, 25, -75, 1099, -4340, 79064, -382060, 8550916, -48306984, 1303568760, -8346754416, 266955481584, -1894529909376, 70785236377728, -547468189825536, 23610353987137536, -196402650598402560, 9679304091074250240, -85687212859582878720, 4785340778000524477440

Offset: 3

Author

Seiichi Manyama, Sep 26 2021

Keywords

• sign

Crossrefs

Column k=3 of A190782.

Cf. A000399, A000454, A054651, A081051, A028340, A347987, A348063, A348065, A348068.

Programs

PARI

a(n) = n!*polcoef(sum(k=3, n, binomial(x, k)), 3);

PARI

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^3/(6*(1-x))))

Python

from sympy.abc import x
from sympy import ff, expand
def A348064(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**3) for k in range(3,n+1)) # Chai Wah Wu, Sep 27 2021

Formula

E.g.f.: (log(1 + x))^3/(6 * (1 - x)).