A361279 Expansion of e.g.f. exp(x * (1+x)^3).

1, 1, 7, 37, 241, 1981, 17551, 171697, 1860097, 21609721, 268697431, 3566446621, 50060084977, 740156116597, 11496472967071, 186824483634601, 3167058238988161, 55882288483846897, 1023891003620741287, 19440027237549627541, 381822392009503555441

Offset

0,3

Links

Winston de Greef, Table of n, a(n) for n = 0..508

Formula

a(n) = n! * Sum_{k=0..n} binomial(3*k,n-k)/k!.

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(3,k-1) * a(n-k)/(n-k)!.

D-finite with recurrence a(n) -a(n-1) +6*(-n+1)*a(n-2) -9*(n-1)*(n-2)*a(n-3) -4*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Mar 13 2023

a(n) ~ 2^(n/2 - 1) * n^(3*n/4) / exp(3*n/4 - 3*n^(3/4)/2^(3/2) - 15*n^(1/2)/64 + n^(1/4)/2^(19/2) + 27/1024) * (1 + 724053*sqrt(2)/(2621440*n^(1/4))). - Vaclav Kotesovec, Nov 11 2023

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^3)))
(PARI) a(n) = n!*sum(k=0, n, binomial(3*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(3, j-1)*v[i-j+1]/(i-j)!)); v;

Crossrefs

Column k=3 of A361277.

Cf. A091695.

Sequence in context: A285846 A102760 A332906 * A096965 A159597 A217723

Adjacent sequences: A361276 A361277 A361278 * A361280 A361281 A361282

Keyword

nonn

Author

Seiichi Manyama, Mar 06 2023

Status

approved