A390061 E.g.f. A(x) satisfies A(x) = exp( x * (1+x^2)^3 * A(x)^2 ).

1, 1, 5, 67, 1089, 23821, 660733, 22101255, 867449537, 39103934041, 1991308574421, 113069588251819, 7083758583603265, 485414045651482533, 36117783340391254349, 2899968818809321621231, 249924002064387424366977, 23011587096024262822463665, 2254436872959357528037779493

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(3*(n-2*k),k)/(n-2*k)!.

E.g.f.: exp( -LambertW(-2*x * (1+x^2)^3)/2 ).

Mathematica

Table[n!*Sum[(2*(n-2*k)+1)^(n-2*k-1)*Binomial[3*(n-2*k), k]/(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 01 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (2*(n-2*k)+1)^(n-2*k-1)*binomial(3*(n-2*k), k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(2*(n-2*k)+1)^(n-2*k-1)* Binomial(3*(n-2*k), k) / Factorial(n-2*k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Nov 02 2025

Crossrefs

Cf. A390058, A390060.

Cf. A389986.

Sequence in context: A316146 A380403 A113265 * A390304 A124435 A123034

Adjacent sequences: A390058 A390059 A390060 * A390062 A390063 A390064

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved