A390058 E.g.f. A(x) satisfies A(x) = exp( x * (1+x^2) * A(x)^2 ).

1, 1, 5, 55, 849, 17581, 460573, 14588715, 542441441, 23170145401, 1118234541141, 60183429196639, 3574085334503665, 232170660260976165, 16376729301439699949, 1246589141412338869651, 101852504879688571752897, 8891022166948825282737265, 825829937078565345973350949

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)} (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

a[n_]:=n!*Sum[(2*(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 27 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (2*(n-2*k)+1)^(n-2*k-1)*binomial(n-2*k, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(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, Oct 27 2025

Crossrefs

Cf. A362773, A390059.

Sequence in context: A132865 A294051 A145662 * A362653 A390301 A094418

Adjacent sequences: A390055 A390056 A390057 * A390059 A390060 A390061

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved