A361570 Expansion of e.g.f. exp( (x * (1+x))^2 ).

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

Links

Table of n, a(n) for n=0..21.

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)

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

Cf. A047974, A361571.

Cf. A052887, A361278.

Sequence in context: A169630 A192385 A352281 * A294464 A366618 A330781

Adjacent sequences: A361567 A361568 A361569 * A361571 A361572 A361573

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2023

Status

approved