A347987 a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x,k).

1, 1, 11, -75, 3969, -140595, 7374191, -435638203, 30421321073, -2409092861175, 214562251828275, -21195275581114635, 2301157855016159905, -272330254968023391035, 34894294917147760652775, -4812715265513253499593675, 710922905477027337578759265, -111981455662673544130741177455

Offset: 0

Author

Seiichi Manyama, Sep 23 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..326

Crossrefs

Cf. A054651, A098118, A190782, A347989.

Programs

Mathematica

Table[(2*n)!/n! * SeriesCoefficient[Log[1+x]^n/(1-x), {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, May 25 2025 *)

PARI

a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x, k)), n);

Formula

a(n) = A054651(2*n,n) = A190782(2*n,n).

a(n) = [x^(2*n)] ((2*n)!/n!) * (log(1 + x))^n/(1 - x).

a(n) ~ (-1)^n * c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), where w = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... and c = 0.07543488970038444052522917317552747476381171100725972392415521577... - Vaclav Kotesovec, Sep 27 2021, updated May 27 2025