login
Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).
4

%I #11 Mar 17 2023 08:27:58

%S 1,1,3,15,99,795,7485,80745,981225,13253625,196834995,3185662095,

%T 55770765435,1049572599075,21120725230605,452384160453225,

%U 10272547048388625,246434674107647025,6226347228582355875,165224032352989584975,4593512876411509125075

%N Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).

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

%F From _Vaclav Kotesovec_, Mar 17 2023: (Start)

%F a(n) = (3*n - 2)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).

%F 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)

%t Table[n! * Sum[Binomial[n,2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 17 2023 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(2*(1-x)^2))/(1-x)))

%Y Cf. A002720, A361597.

%Y Cf. A335344.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 16 2023