A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

1, 0, 2, 6, 36, 240, 2280, 27720, 425040, 7862400, 171188640, 4319330400, 125199708480, 4142318019840, 155388782989440, 6557345831836800, 308677784640825600, 16079233115648102400, 920518264903690252800, 57603377545940850624000

Offset: 0

Author

Seiichi Manyama, Sep 17 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..315

Crossrefs

Cf. A354436, A355575.

Cf. A104872, A216507, A356628.

Programs

Mathematica

Join[{1}, Table[n!*Sum[k^(n - 2*k)/k!, {k, 0, n/2}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

PARI

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

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^(2*k)/(k!*(1-k*x)))))

Formula

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

a(n) ~ sqrt(2*Pi) * exp((n - 1/2)/LambertW(exp(2/3)*(2*n - 1)/6) - 2*n) * n^(2*n + 1/2) / (3^(n + 1/2) * sqrt(1 + LambertW(exp(2/3)*(2*n - 1)/6)) * LambertW(exp(2/3)*(2*n - 1)/6)^n). - Vaclav Kotesovec, Oct 30 2022