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A285064 Row sums of Sheffer triangle S2[4,1] = A285061.

%I A285064 #27 Jun 27 2022 03:08:06
%S A285064 1,5,41,429,5329,75989,1215481,21453693,412820385,8579772325,
%T A285064 191166679497,4538638641997,114238219541617,3035305413035125,
%U A285064 84819458105387417,2484842038066995485,76101249873390595905,2430497813260105226053,80769536433102942870377,2787318255464814752951533
%N A285064 Row sums of Sheffer triangle S2[4,1] = A285061.
%C A285064 See A285061 for details. These are generalized Bell numbers (A000110) because A285061 is a generalized Stirling2 triangle.
%C A285064 For the alternating row sums of A285061 see A285065.
%F A285064 a(n) = Sum_{m=0..n} A285061(n, m), n >= 0.
%F A285064 E.g.f.: exp(x)*exp(exp(4*x) - 1).
%F A285064 a(n) = (1/e)*Sum_{m>=0} (1/m!)*(1+4*m)^n, n >= 0. (Dobiński type formula from the A285061(n,m) sum formula, after interchange of summations).
%F A285064 a(n) = Sum_{k=0..n} binomial(n, k)*A000110(k)*4^k, n >= 0. From the _Vaclav Kotesovec_ program. This follows from the S2[4,1] formula in terms of Stirling2. - _Wolfdieter Lang_, Apr 24 2017
%F A285064 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k). - _Ilya Gutkovskiy_, Jun 21 2022
%F A285064 a(n) ~ Bell(n) * (4 + LambertW(n)/n)^n. - _Vaclav Kotesovec_, Jun 22 2022
%F A285064 a(n) ~ 4^n * n^(n + 1/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/4)). - _Vaclav Kotesovec_, Jun 27 2022
%t A285064 Table[Sum[Binomial[n, k]*BellB[k]*4^k, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 19 2017 *)
%o A285064 (Python)
%o A285064 from sympy import binomial, bell
%o A285064 def a(n): return sum([binomial(n, k)*bell(k)*4**k for k in range(n + 1)]) # _Indranil Ghosh_, Apr 19 2017
%Y A285064 Cf. A000110, A285061, A285065.
%Y A285064 Cf. A126390, A284859.
%K A285064 nonn,easy
%O A285064 0,2
%A A285064 _Wolfdieter Lang_, Apr 13 2017