A347989 a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x+k,k).

1, 5, 71, 1665, 54649, 2310945, 119753843, 7353403057, 522289211873, 42137920501677, 3807384320667135, 380929847762489025, 41811136672902061321, 4995760464106519955705, 645541681316043216096315, 89705032647088734873129825, 13340173206548155385625683265, 2114001534402053456524492822485

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..325

Formula

a(n) = (2*n)! * Sum_{k=n..2*n} (2*n+1-k) * |Stirling1(k,n)|/k!.

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

From Vaclav Kotesovec, Sep 23 2021: (Start)

a(n) = [x^n] Gamma(2*n + x + 2) / Gamma(x + 2).

a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), w = -LambertW(-1,-exp(-1/2)/2) and c = 1.5967712192197964362930380385801737624829174112909160160618... (End)

Prog

(PARI) a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x+k, k)), n);
(PARI) a(n) = (2*n)!*sum(k=n, 2*n, (2*n+1-k)*abs(stirling(k, n, 1))/k!);

Crossrefs

Cf. A001706, A001707, A001708, A001709, A008275, A143491, A347987.

Sequence in context: A362159 A371326 A349471 * A193436 A193501 A133990

Adjacent sequences: A347986 A347987 A347988 * A347990 A347991 A347992

Keyword

nonn

Author

Seiichi Manyama, Sep 23 2021

Status

approved