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A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6*k - 1)^n / (6^k * k!).

%I A337042 #9 Jun 28 2022 06:42:23
%S A337042 1,0,6,36,324,3456,43416,618192,9778320,169827840,3210376032,
%T A337042 65540155968,1435094563392,33510354739200,830486180748672,
%U A337042 21756166766173440,600339119317643520,17394883290643709952,527782830161632077312,16727350847049194775552
%N A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6*k - 1)^n / (6^k * k!).
%C A337042 In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - _Vaclav Kotesovec_, Jun 28 2022
%F A337042 G.f. A(x) satisfies: A(x) = (1 - 6*x + x*A(x/(1 - 6*x))) / (1 - 5*x - 6*x^2).
%F A337042 G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 6*j*x/(1 + x)).
%F A337042 E.g.f.: exp((exp(6*x) - 1) / 6 - x).
%F A337042 a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 6^k * a(n-k-1).
%F A337042 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005012(k).
%F A337042 a(n) ~ 6^(n - 1/6) * n^(n - 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n - 1/6)). - _Vaclav Kotesovec_, Jun 26 2022
%t A337042 nmax = 19; CoefficientList[Series[Exp[(Exp[6 x] - 1)/6 - x], {x, 0, nmax}], x] Range[0, nmax]!
%t A337042 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 6^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
%t A337042 Table[Sum[(-1)^(n - k) Binomial[n, k] 6^k BellB[k, 1/6], {k, 0, n}], {n, 0, 19}]
%Y A337042 Cf. A000296, A003578, A005012, A337038, A337039, A337040, A337041, A337043.
%K A337042 nonn
%O A337042 0,3
%A A337042 _Ilya Gutkovskiy_, Aug 12 2020