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, -52219712942449774799616

Offset

5,2

Links

Table of n, a(n) for n=5..24.

Formula

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

a(n) = Sum_{k=0..n} binomial(n,k) * Stirling1(k,5) * (n-k)!. - Ilya Gutkovskiy, Sep 23 2025

Mathematica

CoefficientList[Series[(Log[1 + x])^5/(120 * (1 - x)), {x, 0, 23}], x]*Range[0, 23]! (* Stefano Spezia, Sep 19 2025 *)

Prog

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

Crossrefs

Column k=5 of A190782.

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

Sequence in context: A143167 A201532 A180788 * A241804 A156949 A155624

Adjacent sequences: A348065 A348066 A348067 * A348069 A348070 A348071

Keyword

sign

Author

Seiichi Manyama, Sep 27 2021

Extensions

a(24) from Stefano Spezia, Sep 19 2025

Status

approved