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A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 2)^n / (4^k * k!).

%I A355165 #9 Jun 27 2022 02:58:19
%S A355165 1,3,13,79,601,5339,53861,607527,7560625,102637235,1506225085,
%T A355165 23726435583,398852249097,7120170905995,134408217821205,
%U A355165 2673140092099543,55832167947587425,1221199519275467107,27902127744298836845,664446811342185649583,16457968670922936733113,423242969435491221774907
%N A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 2)^n / (4^k * k!).
%F A355165 E.g.f.: exp(2*x + (exp(4*x) - 1) / 4).
%F A355165 a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^(k-1) * a(n-k).
%F A355165 a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004213(k).
%F A355165 a(n) ~ 2^(2*n+1) * n^(n + 1/2) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 1/2)). - _Vaclav Kotesovec_, Jun 27 2022
%t A355165 nmax = 21; CoefficientList[Series[Exp[2 x + (Exp[4 x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
%t A355165 a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
%t A355165 Table[Sum[Binomial[n, k] 2^(n + k) BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]
%Y A355165 Cf. A003576, A004213, A285064, A355164, A355167.
%K A355165 nonn
%O A355165 0,2
%A A355165 _Ilya Gutkovskiy_, Jun 22 2022