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%I #7 Jun 21 2022 12:46:18
%S 1,2,11,93,1041,14541,243747,4767183,106556373,2679469065,74864397015,
%T 2300883358995,77144051804409,2802027511061325,109604157405491691,
%U 4593512301562215783,205348466229473678301,9753645833118762303249,490530576727430107027839,26040317900991310393061499
%N Expansion of e.g.f. 3 / (4 - 3*x - exp(3*x)).
%F a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 3^(k-1) * a(n-k).
%F a(n) ~ n! / ((1 + LambertW(exp(4))) * ((4 - LambertW(exp(4)))/3)^(n+1)). - _Vaclav Kotesovec_, Jun 19 2022
%t nmax = 19; CoefficientList[Series[3/(4 - 3 x - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%Y Cf. A003575, A006155, A032033, A255927, A343673, A355110, A355112, A355113, A355114.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 19 2022
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