A390013 E.g.f. A(x) satisfies A(x) = exp( x * (1-x^2) * A(x) ).

1, 1, 3, 10, 53, 336, 2887, 30304, 395817, 6102784, 109219211, 2206139904, 49543830877, 1220029069312, 32645494819407, 942105767366656, 29155253491158353, 962920464001990656, 33804330124398965011, 1257010041289297494016, 49356978868692732793221

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

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

Mathematica

a[n_]:=n!*Sum[(-1)^k*(n-2*k+1)^(n-2*k-1)*Binomial[n-2*k, k]/(n-2*k)!, {k, 0, Floor[n/3]}]; Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 23 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (-1)^k*(n-2*k+1)^(n-2*k-1)*binomial(n-2*k, k)/(n-2*k)!);
(Magma) a := func< n | Factorial(n) * &+[ (-1)^k * (n-2*k + 1)^(n-2*k - 1) * Binomial(n-2*k, k) / Factorial(n-2*k) : k in [0..Floor(n/3)] ] >;
[ a(n) : n in [0..25] ]; // Vincenzo Librandi, Oct 23 2025

Crossrefs

Cf. A389104, A390014.

Cf. A376577.

Sequence in context: A133148 A189815 A143599 * A264409 A199202 A135829

Adjacent sequences: A390010 A390011 A390012 * A390014 A390015 A390016

Keyword

nonn

Author

Seiichi Manyama, Oct 22 2025

Status

approved