A370669 Expansion of Sum_{k>=0} k! * ( x/(1+x^2) )^k.

1, 1, 2, 5, 20, 103, 630, 4475, 36232, 329341, 3320890, 36787889, 444125628, 5803850515, 81625106990, 1229298774647, 19738870726160, 336627732586105, 6076590994501938, 115752541255203869, 2320456607696181220, 48833227436258924671, 1076420625931284514342

Offset: 0

Author

Seiichi Manyama, Feb 25 2024

Keywords

• nonn

Crossrefs

Cf. A000255, A370670.

Cf. A177249, A212580, A276080.

Programs

PARI

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

PARI

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

Formula

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

a(n) = n*a(n-1) + (n-4)*a(n-3) + a(n-4) for n > 4.