%I #6 Dec 05 2023 04:28:57
%S 1,2,8,36,196,1196,8116,60108,481140,4126540,37671540,364068172,
%T 3707910772,39645022540,443540780660,5177560304972,62903920321140,
%U 793654042136908,10378403752717940,140413475790402892,1962339063781284468,28287778534523140428,420059992540347885172
%N Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).
%F G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - x)) / (1 - x) ).
%F a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-2)^n / k!.
%F a(0) = 1; a(n) = -2 * 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) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = -2 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, A126617, A194689, A367888, A367891, A367919, A367921.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 04 2023