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A365863 a(0) = 1; thereafter a(n) = n*Sum_{k = 0..n-1} binomial(n, k)*(-1)^(1+n+k)*a(k).

%I A365863 #45 Nov 13 2023 04:00:09
%S A365863 1,1,2,12,156,3380,108930,4876242,289111032,21916777752,2067208751790,
%T A365863 237380181141950,32601704893973556,5276471519805880836,
%U A365863 993835167745129599162,215520207875112312124890,53311353846240820033325040,14919977169758349265112350256,4690364757880376663319746737926
%N A365863 a(0) = 1; thereafter a(n) = n*Sum_{k = 0..n-1} binomial(n, k)*(-1)^(1+n+k)*a(k).
%C A365863 Let P_k(x) be the polynomial of order k which satisfies a(m) = P_k(m) for m = 0..k, then a(k+1) = k * P_k(k+1).
%C A365863 This sequence is a member of a family of sequences with related properties. Here are some examples:
%C A365863 With b(k+1) = 1 + P_k(k+1) we get b(k) = A000079(k).
%C A365863 With b(k+1) = 2 + P_k(k+1) we get b(k) = A000225(k).
%C A365863 With b(k+1) = 3 + P_k(k+1) we get b(k) = A033484(k).
%C A365863 With b(k+1) = 2 * P_k(k+1) we get b(k) = A000629(k).
%C A365863 With b(k+1) = 1 + 2 * P_k(k+1) we get b(k) = A007047(k).
%C A365863 With b(k+1) = 3 * P_k(k+1) we get b(k) = A201339(k).
%C A365863 With b(k+1) = 5 * P_k(k+1) we get b(k) = A201365(k).
%C A365863 With b(k+1) = -1 * P_k(k+1) we get b(k) = A000670(k)*(-1)^k.
%C A365863 With b(k+1) = -2 * P_k(k+1) we get b(k) = A004123(k+1)*(-1)^k.
%C A365863 With b(k+1) = -3 * P_k(k+1) we get b(k) = A032033(k)*(-1)^k.
%C A365863 With b(k+1) = -4 * P_k(k+1) we get b(k) = A094417(k)*(-1)^k.
%C A365863 With b(k+1) = -m * P_k(k+1) we get b(k) = Bo(m, k)*(-1)^k, Bo(m, k) are Generalized ordered Bell numbers.
%F A365863 a(n) ~ c * n^(2*n + 1/2) / exp(2*n), where c = 2.9711739498821842863440481701659942323709511474486414... - _Vaclav Kotesovec_, Nov 12 2023
%t A365863 a[n_] := a[n] = If[n == 0, 1, n*Sum[Binomial[n, k]*(-1)^(1 + n + k)*a[k], {k, 0, n - 1}]]; Table[a[n], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 12 2023 *)
%o A365863 (PARI) a(n) = if(n == 0, 1,sum(k = 0,n-1, n*binomial(n, k)*(-1)^(1+n+k)*a(k)))
%Y A365863 Cf. A000079, A000225, A000629, A000670.
%Y A365863 Cf. A004123, A007047, A032033, A033484.
%Y A365863 Cf. A094417, A201339, A201365, A367308.
%K A365863 nonn
%O A365863 0,3
%A A365863 _Thomas Scheuerle_, Nov 09 2023