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A361570
Expansion of e.g.f. exp( (x * (1+x))^2 ).
1
1, 0, 2, 12, 36, 240, 2280, 15120, 122640, 1330560, 13335840, 136382400, 1657212480, 20860519680, 262278656640, 3585207225600, 52249374777600, 772773281280000, 11907924610982400, 193962388523904000, 3253343368231756800, 56051640629816832000
OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) ~ 2^(n/2 - 1) * exp(1/64 - 3*n^(1/4)/2^(13/2) - sqrt(n)/16 + n^(3/4)/sqrt(2) - 3*n/4) * n^(3*n/4).
a(n) = 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 4*(n-3)*(n-2)*(n-1)*a(n-4). (End)
MATHEMATICA
With[{nn=30}, CoefficientList[Series[Exp[(x(1+x))^2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 18 2024 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(1+x))^2)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*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