A320502 - OEIS

%I #25 Sep 08 2022 08:46:23

%S 1,1,5,50,842,21644,792676,39297600,2536525008,206794669104,

%T 20785423425264,2525457805492896,364910211591903072,

%U 61847041340997089280,12151693924459271926272,2739901558132307387349504,702704348810821821056454144,203409730893592265642619623424

%N a(n) = Sum_{k=0..n} (k!)^2 * abs(Stirling1(n,k)).

%H Seiichi Manyama, <a href="/A320502/b320502.txt">Table of n, a(n) for n = 0..253</a>

%F a(n) ~ exp(1/2) * (n!)^2.

%F E.g.f.: Sum_{k>=0} k! * (-log(1-x))^k. - _Seiichi Manyama_, Apr 22 2022

%t Table[Sum[Abs[StirlingS1[n, k]]*k!^2, {k, 0, n}], {n, 0, 20}]

%o (PARI) a(n) = sum(k=0, n, k!^2*abs(stirling(n, k, 1))); \\ _Michel Marcus_, Oct 14 2018

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, k!*(-log(1-x))^k))) \\ _Seiichi Manyama_, Apr 22 2022

%o (Magma) [(&+[Abs(StirlingFirst(n,k))*(Factorial(k))^2: k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Oct 14 2018

%Y Cf. A000670, A007840, A064618, A192554.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Oct 13 2018