A365283 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^2).

1, 1, 2, 12, 120, 1380, 19440, 341040, 7029120, 164762640, 4355769600, 128527439040, 4181332700160, 148633442717760, 5734427199621120, 238676208285715200, 10659325532663808000, 508452777299622355200, 25800664274991135129600

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

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

a(n) ~ 2^(n/2) * (1 + 3*LambertW(2^(1/3)/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(2^(1/3)/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(2^(1/3)/3)^(3*(n+1)/2)). - Vaclav Kotesovec, Nov 08 2023

Mathematica

Join[{1}, Table[n!/(n+1) * Sum[(n-2*k)^k * Binomial[n+1, n-2*k]/k!, {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 08 2023 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+1, n-2*k)/k!)/(n+1);

Crossrefs

Cf. A358064, A365282, A365284.

Cf. A161633, A365287.

Sequence in context: A370876 A329851 A127112 * A003580 A052580 A134095

Adjacent sequences: A365280 A365281 A365282 * A365284 A365285 A365286

Keyword

nonn

Author

Seiichi Manyama, Aug 31 2023

Status

approved