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

1, 1, 2, 12, 156, 3380, 108930, 4876242, 289111032, 21916777752, 2067208751790, 237380181141950, 32601704893973556, 5276471519805880836, 993835167745129599162, 215520207875112312124890, 53311353846240820033325040, 14919977169758349265112350256, 4690364757880376663319746737926

Offset: 0

Author

Thomas Scheuerle, Nov 09 2023

Keywords

• nonn

Comments

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).

This sequence is a member of a family of sequences with related properties. Here are some examples:

With b(k+1) = 1 + P_k(k+1) we get b(k) = A000079(k).

With b(k+1) = 2 + P_k(k+1) we get b(k) = A000225(k).

With b(k+1) = 3 + P_k(k+1) we get b(k) = A033484(k).

With b(k+1) = 2 * P_k(k+1) we get b(k) = A000629(k).

With b(k+1) = 1 + 2 * P_k(k+1) we get b(k) = A007047(k).

With b(k+1) = 3 * P_k(k+1) we get b(k) = A201339(k).

With b(k+1) = 5 * P_k(k+1) we get b(k) = A201365(k).

With b(k+1) = -1 * P_k(k+1) we get b(k) = A000670(k)*(-1)^k.

With b(k+1) = -2 * P_k(k+1) we get b(k) = A004123(k+1)*(-1)^k.

With b(k+1) = -3 * P_k(k+1) we get b(k) = A032033(k)*(-1)^k.

With b(k+1) = -4 * P_k(k+1) we get b(k) = A094417(k)*(-1)^k.

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.

Crossrefs

Cf. A000079, A000225, A000629, A000670.

Cf. A004123, A007047, A032033, A033484.

Cf. A094417, A201339, A201365, A367308.

Programs

Mathematica

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 *)

PARI

a(n) = if(n == 0, 1,sum(k = 0,n-1, n*binomial(n, k)*(-1)^(1+n+k)*a(k)))

Formula

a(n) ~ c * n^(2*n + 1/2) / exp(2*n), where c = 2.9711739498821842863440481701659942323709511474486414... - Vaclav Kotesovec, Nov 12 2023