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A361567
Expansion of e.g.f. exp(x^2/2 * (1+x)^2).
5
1, 0, 1, 6, 15, 60, 555, 3150, 17745, 158760, 1399545, 10914750, 102920895, 1104323220, 11249313075, 119330961750, 1426411411425, 17429852840400, 213417453474225, 2791671804271350, 38524272522310575, 537569719902715500, 7732658753799054075
OFFSET
0,4
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/(2^k * k!).
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = (n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n/4 - 1) * exp(1/128 - 3*2^(-29/4)*n^(1/4) - sqrt(n/2)/16 + 2^(-3/4)*n^(3/4) - 3*n/4) * n^(3*n/4). (End)
MATHEMATICA
Table[n! * Sum[Binomial[2*k, n-2*k]/(2^k * k!), {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(1+x)^2)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 16 2023
STATUS
approved