A390303 E.g.f. A(x) satisfies A(x) = exp( x/(1-x^2)^2 * A(x)^2 ).

1, 1, 5, 61, 969, 20881, 564253, 18424365, 705953489, 31068175681, 1544674935861, 85639163765629, 5238842579456665, 350543461952232849, 25469377828007720909, 1996943627149749091501, 168058919566729042456737, 15110749051632106655956225, 1445663161469423481333931621

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-3*k-1,k)/(n-2*k)!.

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

Mathematica

Table[n!*Sum[(2*(n-2*k)+1)^(n-2*k-1)*Binomial[2*n-3*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(2*n-3*k-1, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(2*(n-2*k)+1)^(n-2*k-1)* Binomial(2*n-3*k-1, k) / Factorial(n-2*k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Nov 01 2025

Crossrefs

Cf. A387900, A390094, A390301, A390304.

Cf. A362776, A390270.

Sequence in context: A162167 A134282 A390060 * A146760 A294024 A083082

Adjacent sequences: A390300 A390301 A390302 * A390304 A390305 A390306

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2025

Status

approved