A361570 - OEIS

%I #13 Mar 25 2023 08:54:03

%S 1,0,2,12,36,240,2280,15120,122640,1330560,13335840,136382400,

%T 1657212480,20860519680,262278656640,3585207225600,52249374777600,

%U 772773281280000,11907924610982400,193962388523904000,3253343368231756800,56051640629816832000

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

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

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.

%F From _Vaclav Kotesovec_, Mar 25 2023: (Start)

%F 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).

%F 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)

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

%o (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;

%Y Cf. A047974, A361571.

%Y Cf. A052887, A361278.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 16 2023