A361278 Expansion of e.g.f. exp(x * (1+x)^2).

1, 1, 5, 19, 97, 581, 3661, 26335, 202049, 1659817, 14621941, 135567851, 1326672865, 13624218349, 146056961597, 1633376573431, 18980051829121, 228677164878545, 2852155973178469, 36740599423566787, 488127224550517601, 6678832987859315221

Offset: 0

Author

Seiichi Manyama, Mar 06 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..566

Crossrefs

Column k=2 of A361277.

Cf. A082579.

Programs

Maple

A361278 := proc(n)
    n!*add(binomial(2*k,n-k)/k!,k=0..n) ;
end proc:
seq(A361278(n),n=0..60) ; #R. J. Mathar, Mar 12 2023

PARI

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

PARI

a(n) = n!*sum(k=0, n, binomial(2*k, n-k)/k!);

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(2, j-1)*v[i-j+1]/(i-j)!)); v;

Formula

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

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

D-finite with recurrence a(n) -a(n-1) +4*(-n+1)*a(n-2) -3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Mar 12 2023

a(n) ~ 3^(n/3 - 1/2) * n^(2*n/3) / exp(2*n/3 - 2*3^(-2/3)*n^(2/3) - 3^(-7/3)*n^(1/3) + 4/81) * (1 + 953*3^(1/3)/(4374*n^(1/3)) - 2051059*3^(2/3)/(191318760*n^(2/3))). - Vaclav Kotesovec, Nov 11 2023