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A368718 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^5 / k!.

%I A368718 #23 Jul 18 2025 12:01:02
%S A368718 0,1,30,153,412,1065,1386,7105,-24072,275697,-2656970,29387721,
%T A368718 -352403820,4581620953,-64142155518,962133092145,-15394128425744,
%U A368718 261700184657505,-4710603321945522,89501463119441017,-1790029262385620340,37590614510102111241
%N A368718 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^5 / k!.
%C A368718 In general, for m >=0, Sum_{k=0..n} (-1)^(n-k) * k^m / k! ~ A000587(m) * (-1)^n * exp(-1). - _Vaclav Kotesovec_, Jul 18 2025
%H A368718 Robert Israel, <a href="/A368718/b368718.txt">Table of n, a(n) for n = 0..448</a>
%H A368718 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A368718 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A368718 a(0) = 0; a(n) = -n*a(n-1) + n^5.
%F A368718 E.g.f.: B_5(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
%F A368718 a(n) ~ -2*(-1)^n * exp(-1) * n!. - _Vaclav Kotesovec_, Jul 18 2025
%p A368718 f:= proc(n) option remember;
%p A368718   - n*procname(n-1)+n^5
%p A368718 end proc:
%p A368718 f(0):= 0:
%p A368718 seq(f(i),i=0..21); # _Robert Israel_, May 13 2025
%t A368718 Table[-5*n + 3*n^3 + n^4 - 2*(-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 18 2025 *)
%o A368718 (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 5, stirling(5, k, 2)*x^k)*exp(x)/(1+x))))
%Y A368718 Column k=5 of A368724.
%Y A368718 Cf. A048993, A368587, A368719.
%K A368718 sign
%O A368718 0,3
%A A368718 _Seiichi Manyama_, Jan 04 2024