A392820 - OEIS

%I #13 Jan 25 2026 08:19:12

%S 1,0,0,6,24,70,900,10514,88368,969966,14565660,210128842,3101829192,

%T 53657039462,1011396628788,19614301352130,407298714601056,

%U 9155878501901278,216531181241189964,5365383438167708858,140625865483389408120,3879395530456330528854

%N Expansion of e.g.f. 1/(1 - x*(exp(x) - 1)^2).

%H Vincenzo Librandi, <a href="/A392820/b392820.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = n! * Sum_{k=0..floor(n/3)} (2*k)! * Stirling2(n-k,2*k)/(n-k)!.

%t Table[n!*Sum[(2*k)!*Abs[StirlingS2[n-k,2*k]/(n-k)!],{k,0,Floor[n/3]}],{n,0,23}] (* _Vincenzo Librandi_, Jan 25 2026 *)

%o (PARI) a(n) = n!*sum(k=0, n\3, (2*k)!*stirling(n-k, 2*k, 2)/(n-k)!);

%o (Magma) [Factorial(n)* &+[Factorial(2*k)*Abs(StirlingSecond(n-k, 2*k))/Factorial(n-k) : k in [0..Floor(n/3)] ] : n in [0..23] ]; // _Vincenzo Librandi_, Jan 25 2026

%Y Column k=2 of A392817.

%K nonn,easy

%O 0,4

%A _Seiichi Manyama_, Jan 24 2026