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%I #8 Dec 07 2023 08:23:37
%S 1,5,33,261,2369,24069,269153,3272453,42858113,600181765,8933677729,
%T 140645797125,2332169258945,40586333768197,738998405168609,
%U 14040304543111941,277678389593341185,5704502830382733317,121500343635119818017,2678407616841000605957,61015572313688043492929
%N Expansion of e.g.f. exp(2*(exp(2*x) - 1) + x).
%F G.f. A(x) satisfies: A(x) = 1 + x * ( A(x) + 4 * A(x/(1 - 2*x)) / (1 - 2*x) ).
%F a(n) = exp(-2) * Sum_{k>=0} 2^k * (2*k+1)^n / k!.
%F a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^(k+1) * a(n-k).
%t nmax = 20; CoefficientList[Series[Exp[2 (Exp[2 x] - 1) + x], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(2*x) - 1) + x))) \\ _Michel Marcus_, Dec 07 2023
%Y Cf. A000110, A035009, A126390, A308543, A367945.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 05 2023