OFFSET
0,2
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.
From Vaclav Kotesovec, Feb 04 2026: (Start)
a(n) = 15*(n-2)*a(n-1) - 5*(18*n^2 - 90*n + 115)*a(n-2) + (270*n^3 - 2430*n^2 + 7335*n - 7417)*a(n-3) - 135*(n-4)*(n-3)*(3*n^2 - 21*n + 37)*a(n-4) + 27*(n-5)*(n-4)*(n-3)*(3*n - 13)*(3*n - 11)*a(n-5).
a(n) ~ 2^(3/10) * 3^(n + 1/5) * n^(n - 3/10) / (sqrt(5) * exp(n + 1 - 5*2^(-2/5)*3^(-3/5)*n^(2/5))) * (1 + 2^(1/5)*3^(-6/5)/n^(1/5)). (End)
MATHEMATICA
CoefficientList[Series[Exp[ 1/(1-3*x)^(2/3) - 1], {x, 0, 18}], x]Range[0, 18]! (* Stefano Spezia, Mar 31 2025 *)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-3*x)^(2/3)-1)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 16 2025
STATUS
approved