A365284 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^3).

1, 1, 2, 12, 144, 1980, 31680, 630840, 15093120, 411883920, 12607660800, 430740858240, 16265744732160, 671629503504960, 30093198326231040, 1454898560062147200, 75503612563771392000, 4186035286381024876800, 246916968958719605145600

Offset

0,3

Links

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

Formula

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

a(n) ~ sqrt((1 + 2*r^2*s^3) / (12*r^2*s + 9*r^4*s^4)) * n^(n-1) / (exp(n) * r^n), where s = 1.766482823850997284176450269002863328615073785089684545740773169... is the root of the equation 3*(s-1)*LambertW(2*s*(s-1)^2) = 2 and r = 1/sqrt(3*s^3*(s-1)) = 0.280882078734447087396397749882018030987007964077248... - Vaclav Kotesovec, Mar 10 2024

Mathematica

Join[{1}, Table[n! * Sum[(n - 2*k)^k*Binomial[n + k + 1, n - 2*k]/((n + k + 1)*k!), {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vaclav Kotesovec, Mar 10 2024 *)

Prog

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

Crossrefs

Cf. A358064, A365282, A365283.

Sequence in context: A334174 A372993 A067601 * A052740 A227462 A262241

Adjacent sequences: A365281 A365282 A365283 * A365285 A365286 A365287

Keyword

nonn

Author

Seiichi Manyama, Aug 31 2023

Status

approved