A356199 a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).

1, 1, 6, 122, 5991, 556152, 84245291, 18956006323, 5940695613628, 2474958812797662, 1323229303771318595, 883245295259143164922, 719968321620942410875645, 703829776430361739799683993, 812798413118207226439408790038, 1094718407894086754989907938078190

Offset: 0

Author

Alois P. Heinz, Jul 29 2022

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..215

Crossrefs

Cf. A000110, A008277, A048993, A052880, A349524, A349525, A349583, A349654, A349655.

Programs

Maple

b:= proc(n, k, m) option remember; `if`(n=0,
     (k*m+1)^(m-1), m*b(n-1, k, m)+b(n-1, k, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..19);

Mathematica

b[n_, k_, m_] := b[n, k, m] = If[n == 0,
   (k*m+1)^(m-1), m*b[n-1, k, m] + b[n-1, k, m+1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)

PARI

a(n) = sum(k=0, n, (n*k+1)^(k-1) * stirling(n, k, 2)); \\ Michel Marcus, Aug 04 2022

Formula

a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).

a(n) = [x^n] Sum_{k>=0} (n*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).

a(n) = n! * [x^n] 1/exp(LambertW((1 - exp(x))*n)/n) for n > 0, a(0) = 1.

a(n) ~ exp(exp(-1)/2) * n^(2*n - 2). - Vaclav Kotesovec, Aug 07 2022