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A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4*k - 1)^n / (4^k * k!).

%I A337040 #7 Jun 26 2022 12:44:48
%S A337040 1,0,4,16,112,896,8384,88320,1032448,13242368,184591360,2773929984,
%T A337040 44641579008,765196926976,13905753980928,266855007453184,
%U A337040 5388980396818432,114172599765827584,2530858142594760704,58556990344729198592,1411095950792925904896,35347148031264582270976
%N A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4*k - 1)^n / (4^k * k!).
%F A337040 G.f. A(x) satisfies: A(x) = (1 - 4*x + x*A(x/(1 - 4*x))) / (1 - 3*x - 4*x^2).
%F A337040 G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 4*j*x/(1 + x)).
%F A337040 E.g.f.: exp((exp(4*x) - 1) / 4 - x).
%F A337040 a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 4^k * a(n-k-1).
%F A337040 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004213(k).
%F A337040 a(n) ~ 4^(n - 1/4) * n^(n - 1/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n - 1/4)). - _Vaclav Kotesovec_, Jun 26 2022
%t A337040 nmax = 21; CoefficientList[Series[Exp[(Exp[4 x] - 1)/4 - x], {x, 0, nmax}], x] Range[0, nmax]!
%t A337040 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 4^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 21}]
%t A337040 Table[Sum[(-1)^(n - k) Binomial[n, k] 4^k BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]
%Y A337040 Cf. A000296, A003576, A004213, A337038, A337039, A337041, A337042, A337043.
%K A337040 nonn
%O A337040 0,3
%A A337040 _Ilya Gutkovskiy_, Aug 12 2020