OFFSET
0,3
FORMULA
a(0) = 1; a(n) = Sum_{k=1..n} (3^(k-1) - 1) * binomial(n-1,k-1) * a(n-k).
a(n) ~ exp(exp(3*r)/3 - exp(r) + 2/3 - n) * (n/r)^(n + 1/2) / sqrt((1 + 3*r)*exp(3*r) - (1 + r)*exp(r)), where r = LambertW(3*n)/3 - 1/(1 + 3/LambertW(3*n) - 3^(5/3) * n^(2/3) * (1 + LambertW(3*n)) / LambertW(3*n)^(5/3)). - Vaclav Kotesovec, Jul 05 2022
a(n) ~ (3*n/LambertW(3*n))^n * exp(n/LambertW(3*n) - (n/(LambertW(3*n)/3))^(1/3) - n + 2/3) / sqrt(1 + LambertW(3*n)). - Vaclav Kotesovec, Jul 10 2022
MATHEMATICA
nmax = 20; CoefficientList[Series[Exp[Exp[3*x]/3 - Exp[x] + 2/3], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 30 2022 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp(3*x)/3-exp(x)+2/3)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^(j-1)-1)*binomial(i-1, j-1)*v[i-j+1])); v;
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 30 2022
STATUS
approved