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%I #13 Jul 10 2022 07:48:08
%S 1,0,2,8,38,240,1782,14728,134598,1352800,14800502,174593848,
%T 2205456838,29676417680,423455081142,6381678299368,101217742764358,
%U 1684357485887680,29328589792496502,533062885681064088,10091434399407455558,198592474864415055600
%N Expansion of e.g.f. exp(exp(3*x)/3 - exp(x) + 2/3).
%F a(0) = 1; a(n) = Sum_{k=1..n} (3^(k-1) - 1) * binomial(n-1,k-1) * a(n-k).
%F 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
%F 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
%t 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 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp(3*x)/3-exp(x)+2/3)))
%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)-1)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y Cf. A002874, A355378.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jun 30 2022
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