A361568 Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

1, 0, 0, 1, 12, 60, 130, 420, 8400, 101080, 781200, 4435200, 37714600, 607807200, 8660652000, 94007313400, 914497584000, 11566931376000, 198256136478400, 3275456501116800, 46558791351072000, 636647461257808000, 10238792220969312000, 194852563745775936000

Offset

0,5

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k)/(6^k * k!).

a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * binomial(3,k-3) * a(n-k)/(n-k)!.

a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 12*(n-3)*a(n-4) + 15*(n-3)*(n-4)*a(n-5) + 6*(n-3)*(n-4)*(n-5)*a(n-6)). -Seiichi Manyama, Jun 16 2024

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(1+x)^3)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;

Crossrefs

Cf. A047974, A361567, A361569.

Cf. A115055, A361279.

Sequence in context: A120644 A099829 A099830 * A158443 A153792 A229616

Adjacent sequences: A361565 A361566 A361567 * A361569 A361570 A361571

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2023

Status

approved