A058808 - OEIS

A058808

Product{k=1 to n}[S(n,k)], where S(n,k) is a Stirling number of the second kind. (S(n,k) = number of ways of partitioning a set of n elements into k nonempty subsets.)

%I #25 Feb 27 2021 07:38:02

%S 1,1,3,42,3750,2720250,19512927000,1631977354072800,

%T 1833446251541145780000,31323109023670061678062500000,

%U 9087660958278168844264470405352500000

%N Product{k=1 to n}[S(n,k)], where S(n,k) is a Stirling number of the second kind. (S(n,k) = number of ways of partitioning a set of n elements into k nonempty subsets.)

%H Vaclav Kotesovec, <a href="/A058808/b058808.txt">Table of n, a(n) for n = 1..40</a>

%F log(a(n)) ~ n^2 * (log(n) + gamma - 3/2) / 2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 27 2021

%e a(4) = S(4,1)*S(4,2)*S(4,3)*S(4,4) = 1*7*6*1 = 42.

%p a:=n->mul(stirling2(n, k), k=1..n): seq(a(n), n=1..12); # _Zerinvary Lajos_, Jun 28 2007

%t Table[Product[StirlingS2[n, k], {k, 1, n}], {n, 1, 12}] (* _Vaclav Kotesovec_, Feb 26 2021 *)

%o (PARI) a(n) = prod(k=1, n, stirling(n, k, 2)); \\ _Michel Marcus_, Dec 12 2015

%Y Cf. A058807, A294373.

%K easy,nonn

%O 1,3

%A _Leroy Quet_, Jan 02 2001