A361571 Expansion of e.g.f. exp( (x * (1+x))^3 ).

1, 0, 0, 6, 72, 360, 1080, 15120, 302400, 3689280, 32659200, 359251200, 6965481600, 133880947200, 2070484416000, 30305353478400, 559684629504000, 12582442768896000, 271843009108070400, 5401042458152140800, 111578968350001152000, 2657164887872022528000

Offset: 0

Author

Seiichi Manyama, Mar 16 2023

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..476

Crossrefs

Cf. A047974, A361570.

Cf. A361279.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[(x(1+x))^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 06 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(1+x))^3)))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;

Formula

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

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