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A351933
Expansion of e.g.f. exp(x / (1 - x^2/2)).
3
1, 1, 1, 4, 13, 61, 331, 1996, 14449, 109873, 971821, 8995636, 93329941, 1018571269, 12110589583, 151955795356, 2037757374241, 28837620752161, 430834395468889, 6777014821152868, 111663525724783741, 1930478057636642221, 34781068833200111731
OFFSET
0,4
FORMULA
a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1)!/2^k * binomial(n-1,2*k) * a(n-1-2*k) for n > 2.
a(n) ~ n^(n - 1/4) / (2^(n/2 + 5/8) * exp(n - 2^(3/4)*sqrt(n))). - Vaclav Kotesovec, Mar 03 2022
Conjecture D-finite with recurrence +4*a(n) -4*a(n-1) -4*(n-1)*(n-2)*a(n-2) -2*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 09 2022
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k-1,k)/(2^k * (n-2*k)!). - Seiichi Manyama, Jun 08 2024
MATHEMATICA
m = 22; Range[0, m]! * CoefficientList[Series[Exp[x / (1 - x^2/2)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
PROG
(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^2/2))))
(PARI) a(n) = if(n<3, 1, sum(k=0, (n-1)\2, (2*k+1)!/2^k*binomial(n-1, 2*k)*a(n-1-2*k)));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 26 2022
STATUS
approved