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

%I A356199 #22 Feb 14 2023 07:41:19
%S A356199 1,1,6,122,5991,556152,84245291,18956006323,5940695613628,
%T A356199 2474958812797662,1323229303771318595,883245295259143164922,
%U A356199 719968321620942410875645,703829776430361739799683993,812798413118207226439408790038,1094718407894086754989907938078190
%N A356199 a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
%H A356199 Alois P. Heinz, <a href="/A356199/b356199.txt">Table of n, a(n) for n = 0..215</a>
%F A356199 a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
%F A356199 a(n) = [x^n] Sum_{k>=0} (n*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
%F A356199 a(n) = n! * [x^n] 1/exp(LambertW((1 - exp(x))*n)/n) for n > 0, a(0) = 1.
%F A356199 a(n) ~ exp(exp(-1)/2) * n^(2*n - 2). - _Vaclav Kotesovec_, Aug 07 2022
%p A356199 b:= proc(n, k, m) option remember; `if`(n=0,
%p A356199      (k*m+1)^(m-1), m*b(n-1, k, m)+b(n-1, k, m+1))
%p A356199     end:
%p A356199 a:= n-> b(n$2, 0):
%p A356199 seq(a(n), n=0..19);
%t A356199 b[n_, k_, m_] := b[n, k, m] = If[n == 0,
%t A356199    (k*m+1)^(m-1), m*b[n-1, k, m] + b[n-1, k, m+1]];
%t A356199 a[n_] := b[n, n, 0];
%t A356199 Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Feb 14 2023, after _Alois P. Heinz_ *)
%o A356199 (PARI) a(n) = sum(k=0, n, (n*k+1)^(k-1) * stirling(n, k, 2)); \\ _Michel Marcus_, Aug 04 2022
%Y A356199 Cf. A000110, A008277, A048993, A052880, A349524, A349525, A349583, A349654, A349655.
%K A356199 nonn
%O A356199 0,3
%A A356199 _Alois P. Heinz_, Jul 29 2022