A390649 E.g.f. A(x) satisfies A(x) = 1 + x^3*exp(x)*A(x)^3.

1, 0, 0, 6, 24, 60, 2280, 30450, 242256, 5806584, 137895120, 2187441630, 52338909480, 1622613359796, 40478541145224, 1140057843727530, 40943277266414880, 1381413912597093360, 47563489121700469536, 1928848981483349781174, 80049349213837886368440

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k) * A001764(k)/(n-3*k)!.

Mathematica

Join[{1}, Table[Factorial[n]*Sum[k^(n-3*k)*Binomial[3*k, k]/((2*k+1)*Factorial[n-3*k]), {k, 0, Floor[n/3]}], {n, 1, 20}]] (* Vincenzo Librandi, Dec 25 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(3*k, k)/((2*k+1)*(n-3*k)!));
(Magma) [Factorial(n)*&+[k^(n-3*k)*Binomial(3*k, k)/((2*k+1)*Factorial(n-3*k)): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 25 2025

Crossrefs

Cf. A358081, A390648.

Cf. A001764, A390611.

Sequence in context: A362703 A390648 A371046 * A371019 A370985 A101854

Adjacent sequences: A390646 A390647 A390648 * A390650 A390651 A390652

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 13 2025

Status

approved