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%I #10 Jul 04 2020 03:07:57
%S 1,3,7,11,7,-5,23,75,-281,-101,4663,-14229,-41721,532667,-1464489,
%T -8840053,103689511,-313202725,-2348557705,32041266859,-127039882425,
%U -762423051013,14393151011735,-81523161874741,-236027974047897,8564406463119387
%N Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).
%F a(n) = exp(2) * (-1)^n * Sum_{k>=0} (-2)^k * (k - 1)^n / k!.
%F a(0) = 1; a(n) = a(n-1) + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).
%t nmax = 25; CoefficientList[Series[Exp[2 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]
%o (PARI) my(N=33, x='x+O('x^N)); Vec(serlaplace(exp(2 * (1 - exp(-x)) + x))) \\ _Joerg Arndt_, Jul 04 2020
%Y Cf. A001861, A035009, A109747, A213170, A335868, A335981, A335982.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Jul 03 2020
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