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%I #5 Dec 05 2023 04:31:26
%S 1,1,5,17,93,505,3269,22657,172461,1407177,12284629,113832273,
%T 1114775869,11487315481,124118143717,1401808691489,16504815145421,
%U 202101235848297,2568312461002741,33808677627863537,460227870278020957,6468672644291075001,93745096205219336709
%N Expansion of e.g.f. exp(4*(exp(x) - 1) - 3*x).
%F G.f. A(x) satisfies: A(x) = 1 - x * ( 3 * A(x) - 4 * A(x/(1 - x)) / (1 - x) ).
%F a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-3)^n / k!.
%F a(0) = 1; a(n) = -3 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
%t nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 3 x], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
%Y Cf. A000296, A078944, A346738, A355253, A367890, A367891, A367919, A367920.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 04 2023
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