A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).

1, 0, 11, 5, 304, 364, 15980, 34236, 1368936, 4429656, 173699712, 771653376, 30605906304, 175622947200, 7149130156800, 50800930272000, 2137822335475200, 18241636315507200, 796397873127782400, 7971407298921830400, 361615771356450508800, 4168685961862906982400, 196587429737202833817600

Offset

2,3

Links

Table of n, a(n) for n=2..24.

Formula

a(n) = a(n-1) + (n-1)^2 * a(n-2) + (-1)^n * (n-2)!.

E.g.f.: (log(1 + x))^2/(2 * (1 - x)).

a(n) ~ n! * log(2)^2 / 2 * (1 + (-1)^n*log(n)/(log(2)^2*n)). - Vaclav Kotesovec, Sep 27 2021

Prog

(PARI) a(n) = n!*polcoef(sum(k=2, n, binomial(x, k)), 2);
(PARI) a(n) = if(n<2, 0, a(n-1)+(n-1)^2*a(n-2)+(-1)^n*(n-2)!);
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^2/(2*(1-x))))
(Python)
from sympy.abc import x
from sympy import ff, expand
def A348063(n): return sum(ff(n, n-k)*expand(ff(x, k)).coeff(x**2) for k in range(2, n+1)) # Chai Wah Wu, Sep 27 2021

Crossrefs

Column k=2 of A190782.

Cf. A000399, A028339, A054651, A347987, A348064, A348065, A348068.

Sequence in context: A174103 A038319 A002547 * A090840 A227775 A204011

Adjacent sequences: A348060 A348061 A348062 * A348064 A348065 A348066

Keyword

nonn

Author

Seiichi Manyama, Sep 26 2021

Status

approved