A352302 Expansion of e.g.f. 1/(exp(x) - x^2).

1, -1, 3, -13, 73, -521, 4441, -44185, 502545, -6429169, 91393201, -1429101521, 24378097129, -450504733849, 8965682806809, -191174795868841, 4348171177591201, -105077942935229537, 2688685949077138657, -72618903735812907553, 2064598911185525708601

Offset: 0

Author

Seiichi Manyama, Mar 11 2022

Keywords

• sign

Crossrefs

Cf. A089148, A352303, A352304.

Cf. A346269.

Programs

Mathematica

m = 20; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^2), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^2)))

PARI

b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2);

Formula

a(n) = n * (n-1) * a(n-2) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 1.

a(n) ~ n! * (-1)^n / ((1 + LambertW(1/2)) * 2^(n+3) * LambertW(1/2)^(n+2)). - Vaclav Kotesovec, Mar 12 2022

a(n) = n! * Sum_{k=0..floor(n/2)} (-k-1)^(n-2*k)/(n-2*k)!. - Seiichi Manyama, Aug 21 2024