A387900 E.g.f. A(x) satisfies A(x) = exp( x * (1-x^2) * A(x)^2 ).

1, 1, 5, 43, 609, 11701, 285613, 8439495, 292932545, 11682468361, 526465053621, 26458570730659, 1467305459248225, 89009739625732797, 5863237365379517789, 416790183010259307391, 31801839704719698982017, 2592509679168981991215505, 224878909210578388233989605

Offset

0,3

Links

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (2*(n-2*k)+1)^(n-2*k-1) * binomial(n-2*k,k)/(n-2*k)!.

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

Mathematica

Table[n!*Sum[(-1)^k*(2 (n-2 k)+1)^(n-2 k-1)*Binomial[n-2 k, k]/Factorial[n-2 k], {k, 0, Floor[n/3]}], {n, 0, 18}] (* Vincenzo Librandi, Nov 28 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (-1)^k*(2*(n-2*k)+1)^(n-2*k-1)*binomial(n-2*k, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(-1)^k*(2*(n-2*k)+1)^(n-2*k-1)*Binomial(n-2*k, k) / Factorial(n-2*k) : k in [0..Floor(n/3)]] : n in [0..25] ]; // Vincenzo Librandi, Nov 28 2025

Crossrefs

Cf. A389174, A390011.

Cf. A390013, A390058.

Sequence in context: A005989 A307362 A280776 * A390269 A255895 A160450

Adjacent sequences: A387897 A387898 A387899 * A387901 A387902 A387903

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved