A390060 E.g.f. A(x) satisfies A(x) = exp( x * (1+x^2)^2 * A(x)^2 ).

1, 1, 5, 61, 969, 20641, 557053, 18157245, 694542929, 30516930721, 1514856300021, 83852728961629, 5121425440919065, 342142976749328769, 24819534946205847149, 1942897633265080625821, 163250227357672434002337, 14654995842464488153685185, 1399824959394465719739923941

Offset

0,3

Links

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

Formula

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

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

Mathematica

a[n_]:=n!*Sum[(2*(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, (2*(n-2*k)+1)^(n-2*k-1)*binomial(2*(n-2*k), k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(2*(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. A390058, A390061.

Cf. A387013.

Sequence in context: A012167 A162167 A134282 * A390303 A146760 A294024

Adjacent sequences: A390057 A390058 A390059 * A390061 A390062 A390063

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved