A316747 Stirling transform of (2*n)!.

1, 2, 26, 794, 44810, 4050362, 536119946, 97759687034, 23495075990090, 7197163489723322, 2737224615568742666, 1265459307754418362874, 698926543187678223962570, 454516898016585094157146682, 343753040265700944173260034186, 299168865461564926143049346952314

Offset: 0

Author

Vaclav Kotesovec, Jul 12 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..224
• Eric Weisstein's World of Mathematics, Stirling Transform.

Crossrefs

Cf. A000670, A052841, A064618, A316748.

Programs

Mathematica

Table[Sum[StirlingS2[n, k]*(2*k)!, {k, 0, n}], {n, 0, 20}]

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k)!*(exp(x)-1)^k/k!))) \\ Seiichi Manyama, May 20 2022

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * (2*k)!.

a(n) ~ exp(1/8) * (2*n)!.

a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n - 1/8).

E.g.f.: Sum_{k>=0} (2*k)! * (exp(x) - 1)^k / k!. - Seiichi Manyama, May 20 2022