A390269 E.g.f. A(x) satisfies A(x) = exp( x / (1+x^2) * A(x)^2 ).

1, 1, 5, 43, 609, 11821, 289213, 8557935, 297428225, 11877531481, 535974717141, 26973162315859, 1497899053555105, 90991409137652037, 6002116668673048589, 427258830947813153431, 32646332845112386658817, 2665095228257775382091185, 231500830856971358947972645

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

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

Mathematica

Table[n!*Sum[(-1)^k*(2*(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*(2*(n-2*k)+1)^(n-2*k-1)*binomial(n-k-1, k)/(n-2*k)!);

Crossrefs

Cf. A390058, A390060, A390270.

Cf. A390064, A390271.

Sequence in context: A307362 A280776 A387900 * A255895 A160450 A114604

Adjacent sequences: A390266 A390267 A390268 * A390270 A390271 A390272

Keyword

nonn

Author

Seiichi Manyama, Oct 30 2025

Status

approved