A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(n - 2*k)!.

1, 1, 1, 7, 25, 181, 1561, 12811, 188497, 2071945, 38889361, 620762671, 12917838121, 291278938237, 6667342764265, 194869722610291, 5137978752994081, 177509783765281681, 5610285632192738977, 215195998789004395735, 8228064506323330305721

Offset: 0

Author

Seiichi Manyama, Aug 18 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..366

Crossrefs

Cf. A354436, A356629, A356630.

Cf. A216688, A356632, A358064.

Programs

Mathematica

a[n_] := n! * Sum[(n - 2*k)^k/(n - 2*k)!, {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 19 2022 *)

PARI

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

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^2)))))

Formula

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^2)).

a(n) ~ sqrt(Pi) * exp((n-1)/(2*LambertW(exp(1/3)*(n-1)/3)) - 3*n/2) * n^((3*n + 1)/2) / (sqrt(1 + LambertW(exp(1/3)*(n - 1)/3)) * 3^((n+1)/2) * LambertW(exp(1/3)*(n-1)/3)^(n/2)). - Vaclav Kotesovec, Nov 01 2022