A389818 E.g.f. A(x) satisfies A(x) = (1+x) * exp(x^2 * (1+x) * A(x)).

1, 1, 2, 18, 108, 1020, 12720, 169680, 2770320, 51423120, 1058279040, 24441056640, 618994474560, 17089288856640, 512108910593280, 16520379068793600, 571332729077510400, 21090175678149177600, 827472358078036070400, 34394648421001681612800, 1509824201495251446604800

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..390

Formula

E.g.f.: (1+x) * exp( -LambertW(-x^2 * (1+x)^2) ).

a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * binomial(2*k+1,n-2*k)/k!.

Mathematica

Table[n!*Sum[(k+1)^(k-1)*Binomial[2*k+1, n-2*k]/k!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 01 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*binomial(2*k+1, n-2*k)/k!);
(Magma) [Factorial(n) * &+[(k+1)^(k-1)* Binomial(2*k+1, n-2*k) / Factorial(k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Nov 01 2025

Crossrefs

Cf. A378019, A389821.

Cf. A387968.

Sequence in context: A006043 A112328 A370732 * A389786 A038721 A308700

Adjacent sequences: A389815 A389816 A389817 * A389819 A389820 A389821

Keyword

nonn

Author

Seiichi Manyama, Oct 16 2025

Status

approved