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Expansion of e.g.f. exp(4*(exp(x) - 1) + 2*x).
1

%I #19 Dec 07 2023 08:23:41

%S 1,6,40,292,2308,19580,177044,1696572,17148916,182114972,2024979604,

%T 23506175868,284125820724,3567957972316,46454893734612,

%U 625979771144764,8715626185644916,125200337417147932,1853095248414187796,28225529312569364732,441925530173009732532

%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 = 20; 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, 20}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 2*x))) \\ _Michel Marcus_, Dec 07 2023

%Y Cf. A005493, A035009, A078945, A355247, A367889, A367937.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Dec 05 2023