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A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.

%I A355163 #9 Jun 27 2022 02:50:56
%S A355163 1,7,65,743,9921,150151,2526593,46615783,933072513,20093861895,
%T A355163 462440842177,11310514854375,292627518129985,7976748158144647,
%U A355163 228308400790500097,6840702405678586343,214000748166439723265,6973447420429351808007,236204029044752265931585,8300724166287243795922151
%N A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.
%F A355163 E.g.f.: exp(exp(4*x) + 3 x - 1).
%F A355163 a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
%F A355163 a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * Bell(k).
%F A355163 a(n) ~ Bell(n) * (4 + 3*LambertW(n)/n)^n. - _Vaclav Kotesovec_, Jun 22 2022
%F A355163 a(n) ~ 4^n * n^(n + 3/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 3/4)). - _Vaclav Kotesovec_, Jun 27 2022
%t A355163 nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A355163 a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%t A355163 Table[Sum[Binomial[n, k] 3^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]
%Y A355163 Cf. A000110, A005494, A126390, A284859, A284864, A285064, A355162.
%K A355163 nonn
%O A355163 0,2
%A A355163 _Ilya Gutkovskiy_, Jun 22 2022