A380214 Expansion of e.g.f. exp( 1/(1-3*x)^(2/3) - 1 ).

1, 2, 14, 148, 2076, 36152, 750344, 18055088, 493688976, 15108697632, 511379579104, 18959550197568, 763909806479296, 33227876172374912, 1551519044372535424, 77391560357497815808, 4106518327272819159296, 230931323981550384824832, 13718006864544800838290944

Offset: 0

Author

Seiichi Manyama, Jan 16 2025

Keywords

• nonn

Crossrefs

Cf. A049119, A380257.

Programs

Mathematica

CoefficientList[Series[Exp[ 1/(1-3*x)^(2/3) - 1],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-3*x)^(2/3)-1)))

Formula

a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * |Stirling1(n,k)| * Bell(k).

a(n) = (1/e) * (-3)^n * n! * Sum_{k>=0} binomial(-2*k/3,n)/k!.

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380257.