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%I A367945 #9 Dec 07 2023 08:23:29
%S A367945 1,3,17,115,929,8547,87729,988883,12100929,159331523,2241395537,
%T A367945 33493315379,529089873121,8799587162659,153545747910129,
%U A367945 2802447872764307,53358770299683457,1057354788073681283,21760656533457251985,464240718007022020083,10249389749356980403745
%N A367945 Expansion of e.g.f. exp(2*(exp(2*x) - 1) - x).
%F A367945 G.f. A(x) satisfies: A(x) = 1 - x * ( A(x) - 4 * A(x/(1 - 2*x)) / (1 - 2*x) ).
%F A367945 a(n) = exp(-2) * Sum_{k>=0} 2^k * (2*k-1)^n / k!.
%F A367945 a(0) = 1; a(n) = -a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^(k+1) * a(n-k).
%t A367945 nmax = 20; CoefficientList[Series[Exp[2 (Exp[2 x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
%t A367945 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 A367945 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(2*x) - 1) - x))) \\ _Michel Marcus_, Dec 07 2023
%Y A367945 Cf. A000296, A124311, A217924, A308543, A367946.
%K A367945 nonn
%O A367945 0,2
%A A367945 _Ilya Gutkovskiy_, Dec 05 2023

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