A387916 E.g.f. A(x) satisfies A(x) = exp( x * (1-x^3)^2 * A(x)^2 ).

1, 1, 5, 49, 681, 13441, 336013, 10170945, 361749713, 14782435777, 682548840981, 35145505666129, 1996881007498105, 124105026330958401, 8375357686575105437, 609947786960560209121, 47679639725594001339297, 3982021898392415788834945, 353860491590334378389836837

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Formula

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

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

Mathematica

a[n_]:=n!*Sum[(-1)^k*(2*(n-3*k)+1)^(n-3*k-1)*Binomial[2*(n-3*k), k]/(n-3*k)!, {k, 0, Floor[n/3]}]; Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 25 2025 *)

Prog

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

Crossrefs

Cf. A390015.

Sequence in context: A228511 A116873 A324361 * A390272 A089914 A267220

Adjacent sequences: A387913 A387914 A387915 * A387917 A387918 A387919

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved