A390304 E.g.f. A(x) satisfies A(x) = exp( x/(1-x^2)^3 * A(x)^2 ).

1, 1, 5, 67, 1089, 24181, 671533, 22517055, 885774977, 40013538121, 2041871638581, 116180995535419, 7293739621545985, 500836204666837053, 37342342012252979549, 3004493763244790498551, 259468101897601329598977, 23939834074342139346606865, 2350237219965822224472797413

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-5*k-1,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-5*k-1, 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-5*k-1, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(2*(n-2*k)+1)^(n-2*k-1)* Binomial(3*n-5*k-1, k) / Factorial(n-2*k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Nov 02 2025

Crossrefs

Cf. A387900, A390094, A390301, A390303.

Sequence in context: A380403 A113265 A390061 * A124435 A123034 A382058

Adjacent sequences: A390301 A390302 A390303 * A390305 A390306 A390307

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2025

Status

approved