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A368716 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.

%I A368716 #23 Jul 18 2025 11:22:01
%S A368716 0,1,6,9,28,-15,306,-1799,14904,-133407,1335070,-14684439,176214996,
%T A368716 -2290792751,32071101258,-481066515495,7697064252016,-130850092279359,
%U A368716 2355301661034294,-44750731559644727,895014631192902540,-18795307255050944079
%N A368716 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.
%C A368716 abs(a(n))/n is prime for n = 2, 3, 4, 5, 7, 13, 19, 28, 643 and no others up to n = 2000. - _Robert Israel_, May 13 2025
%H A368716 Robert Israel, <a href="/A368716/b368716.txt">Table of n, a(n) for n = 0..448</a>
%H A368716 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A368716 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A368716 a(0) = 0; a(n) = -n*a(n-1) + n^3.
%F A368716 E.g.f.: B_3(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
%F A368716 a(n) ~ (-1)^n * exp(-1) * n!. - _Vaclav Kotesovec_, Jul 18 2025
%p A368716 f:= proc(n) option remember;
%p A368716   - n*procname(n-1)+n^3
%p A368716 end proc:
%p A368716 f(0):= 0:
%p A368716 seq(f(i),i=0..30); # _Robert Israel_, May 13 2025
%t A368716 Table[n + n^2 + (-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 18 2025 *)
%o A368716 (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 3, stirling(3, k, 2)*x^k)*exp(x)/(1+x))))
%Y A368716 Column k=3 of A368724.
%Y A368716 Cf. A048993, A337001, A368585.
%K A368716 sign
%O A368716 0,3
%A A368716 _Seiichi Manyama_, Jan 04 2024