%I #8 Dec 07 2023 08:23:17
%S 1,1,10,55,487,4654,51463,632125,8536492,125279785,1981246555,
%T 33530245984,603797462677,11513675558701,231539488842610,
%U 4893151984630579,108334206855000739,2505977899186557502,60419653270442268643,1515077412621445514089,39437350309301393464876,1063746973172416765272589
%N Expansion of e.g.f. exp(exp(3*x) - 1 - 2*x).
%F G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - 3*x)) / (1 - 3*x) ).
%F a(n) = exp(-1) * Sum_{k>=0} (3*k-2)^n / k!.
%F a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
%F a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * 3^k * Bell(k).
%t nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
%t Table[Sum[Binomial[n, k] (-2)^(n - k) 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - 1 - 2*x))) \\ _Michel Marcus_, Dec 07 2023
%Y Cf. A000110, A124311, A126617, A247452, A284864, A367785, A367888.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 05 2023
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