%I #13 Jul 05 2022 02:24:19

%S 1,0,1,5,22,115,761,5880,49897,460045,4621366,50385555,590795217,

%T 7389964400,98105330961,1377426850805,20388005470582,317112889169555,

%U 5167636268318921,88001180739368680,1562559584723343417,28871671817796197885,554116841783123679446

%N Expansion of e.g.f. exp(exp(3*x)/3 - exp(2*x)/2 + 1/6).

%F a(0) = 1; a(n) = Sum_{k=1..n} (3^(k-1) - 2^(k-1)) * binomial(n-1,k-1) * a(n-k).

%F a(n) ~ exp(exp(3*r)/3 - exp(2*r)/2 + 1/6 - n) * (n/r)^(n + 1/2) / sqrt((1 + 3*r)*exp(3*r) - (1 + 2*r)*exp(2*r)), where r = LambertW(3*n)/3 - 1/(2 + 3/LambertW(3*n) - 3^(4/3) * n^(1/3) * (1 + LambertW(3*n)) / LambertW(3*n)^(4/3)). - _Vaclav Kotesovec_, Jul 05 2022

%t nmax = 20; CoefficientList[Series[Exp[Exp[3*x]/3 - Exp[2*x]/2 + 1/6], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 30 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp(3*x)/3-exp(2*x)/2+1/6)))

%o (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)-2^(j-1))*binomial(i-1, j-1)*v[i-j+1])); v;

%Y Cf. A355381.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jun 30 2022