A387013 E.g.f. A(x) satisfies A(x) = exp(x * (1+x^2)^2 * A(x)).

1, 1, 3, 28, 269, 3336, 53287, 1008064, 22199193, 559954432, 15899946731, 502178632704, 17470278315877, 663858654275584, 27361365974132751, 1215869626302939136, 57952329108501627953, 2949335923883992743936, 159631304307995442321235, 9156282245864958498439168

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..345

Formula

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

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

Mathematica

a[n_]:=n!*Sum[(n-2*k+1)^(n-2*k-1)*Binomial[2*(n-2*k), k]/(n-2*k)!, {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 26 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k+1)^(n-2*k-1)*binomial(2*(n-2*k), k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(n-2*k+1)^(n-2*k-1) * Binomial(2*(n-2*k), k) / Factorial(n-2*k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Oct 26 2025

Crossrefs

Cf. A376577, A389986.

Cf. A362772, A389987.

Sequence in context: A037588 A307885 A076723 * A390182 A198887 A026114

Adjacent sequences: A387010 A387011 A387012 * A387014 A387015 A387016

Keyword

nonn

Author

Seiichi Manyama, Oct 21 2025

Status

approved