A352309 Expansion of e.g.f. 1/(exp(x) - x^2/2).

1, -1, 2, -7, 31, -171, 1141, -8863, 78653, -785557, 8716861, -106395741, 1416724915, -20436548575, 317477947151, -5284248213091, 93816998697721, -1769737117839849, 35347571931577609, -745232024035027225, 16538641134235561631, -385387334950748244451

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

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

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

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

Mathematica

m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^2/2), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^2/2)))
(PARI) b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m))*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2);

Crossrefs

Cf. A089148, A352310, A352311.

Cf. A352302, A352306.

Sequence in context: A321208 A005977 A199675 * A059037 A245313 A306037

Adjacent sequences: A352306 A352307 A352308 * A352310 A352311 A352312

Keyword

sign

Author

Seiichi Manyama, Mar 11 2022

Status

approved