A390214 E.g.f. A(x) satisfies A(x) = exp(x/(1+x^2) * A(x)).

1, 1, 3, 10, 53, 456, 5047, 65584, 993897, 17352064, 342278891, 7518622464, 182100395677, 4822587937792, 138634240482207, 4299201445943296, 143066151934347473, 5085333563651751936, 192298729746624034771, 7708173980843258085376, 326481664040775374745861

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

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

Mathematica

Table[n!*Sum[(-1)^k*(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 04 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (-1)^k*(n-2*k+1)^(n-2*k-1)*binomial(n-k-1, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(-1)^k * (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 04 2025

Crossrefs

Cf. A335945, A390215.

Cf. A376575.

Sequence in context: A199202 A135829 A071895 * A054422 A074503 A318188

Adjacent sequences: A390211 A390212 A390213 * A390215 A390216 A390217

Keyword

nonn

Author

Seiichi Manyama, Oct 29 2025

Status

approved