A390216 E.g.f. A(x) satisfies A(x) = exp(x/(1+x^3)^2 * A(x)).

1, 1, 3, 16, 77, 576, 5287, 67264, 1033209, 18672544, 381420971, 8668210464, 216775643317, 5916230986576, 175080810934863, 5585041595100496, 191083649523054833, 6980309433990052416, 271185659184540450259, 11165343780742628090176, 485654267817827570333421

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 * (n-3*k+1)^(n-3*k-1) * binomial(2*n-5*k-1,k)/(n-3*k)!.

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

Mathematica

Table[n!*Sum[(-1)^k*(n-3*k+1)^(n-3*k-1)*Binomial[2*n-5*k-1, k]/(n-3*k)!, {k, 0, Floor[n/3]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 04 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (-1)^k*(n-3*k+1)^(n-3*k-1)*binomial(2*n-5*k-1, k)/(n-3*k)!);
(Magma) [Factorial(n) * &+[(-1)^k * (n-3*k+1)^(n-3*k-1)* Binomial(2*n-5*k-1, k) / Factorial(n-3*k) : k in [0..Floor(n/3)]] : n in [0..25] ]; // Vincenzo Librandi, Nov 04 2025

Crossrefs

Cf. A376578, A389987, A390215.

Cf. A390184.

Sequence in context: A053572 A329806 A390015 * A371350 A309915 A343117

Adjacent sequences: A390213 A390214 A390215 * A390217 A390218 A390219

Keyword

nonn

Author

Seiichi Manyama, Oct 29 2025

Status

approved