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A361596
Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).
4
1, 1, 3, 15, 99, 795, 7485, 80745, 981225, 13253625, 196834995, 3185662095, 55770765435, 1049572599075, 21120725230605, 452384160453225, 10272547048388625, 246434674107647025, 6226347228582355875, 165224032352989584975, 4593512876411509125075
OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n,2*k)/(2^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 2)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 3^(-1/2) * exp(1/6 - n^(1/3)/2 + 3*n^(2/3)/2 - n) * n^(n + 1/6) * (1 + 49/(108*n^(1/3)) + 3293/(116640*n^(2/3))). (End)
MATHEMATICA
Table[n! * Sum[Binomial[n, 2*k]/(2^k * k!), {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(2*(1-x)^2))/(1-x)))
CROSSREFS
Cf. A335344.
Sequence in context: A111546 A219359 A152402 * A255806 A226515 A135883
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 16 2023
STATUS
approved