A390301 E.g.f. A(x) satisfies A(x) = exp( x/(1-x^2) * A(x)^2 ).

1, 1, 5, 55, 849, 17701, 464173, 14717235, 547743521, 23418551881, 1131271398261, 60941609727439, 3622478292965425, 235533653451029805, 16629469072027244189, 1267012397537263762891, 103618232946308849804097, 9053648828215599307042705, 841725575703516474238796389

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

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

Mathematica

Table[n!*Sum[(2*(n-2*k)+1)^(n-2*k-1)*Binomial[n-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(n-k-1, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(2*(n-2*k)+1)^(n-2*k-1)* Binomial(n-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, A390303, A390304.

Cf. A361065, A390302.

Cf. A390269.

Sequence in context: A145662 A390058 A362653 * A094418 A365588 A367164

Adjacent sequences: A390298 A390299 A390300 * A390302 A390303 A390304

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2025

Status

approved