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A284859 Row sums of the Sheffer triangle (exp(x), exp(3*x)-1) given in A282629.

%I A284859 #23 Jun 27 2022 03:09:59
%S A284859 1,4,25,199,1876,20257,245017,3266914,47450923,743935375,12497579698,
%T A284859 223619318215,4240423494685,84855613320004,1785410320771933,
%U A284859 39373503608087299,907548770965519660,21810536356271794549,545305573054110017125,14155835044848094831018
%N A284859 Row sums of the Sheffer triangle (exp(x), exp(3*x)-1) given in A282629.
%C A284859 See A282629 for details. These are generalized Bell numbers (A000110) because A282629 is a generalized Stirling2 triangle.
%F A284859 a(n) = Sum_{m=0..n} A282629(n, m).
%F A284859 E.g.f.: exp(x)*exp(exp(3*x) -1).
%F A284859 a(n) = (1/e)*Sum_{m>=0} (1/m!)*(1+3*m)^n, n >= 0. (Dobiński type formula from the A282629(n,m) sum formula, interchanging summations).
%F A284859 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - _Ilya Gutkovskiy_, Jun 21 2022
%F A284859 a(n) ~ Bell(n) * (3 + LambertW(n)/n)^n. - _Vaclav Kotesovec_, Jun 22 2022
%F A284859 a(n) ~ 3^n * n^(n + 1/3) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/3)). - _Vaclav Kotesovec_, Jun 27 2022
%t A284859 T[n_, m_]:= Sum[Binomial[m, k] (-1)^(k - m) (1 + 3k)^n/m!, {k, 0, m}]; Table[Sum[T[n, m], {m, 0, n}], {n, 0, 20}] (* _Indranil Ghosh_, Apr 10 2017 *)
%t A284859 Table[Sum[3^k*Binomial[n,k]*BellB[k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 22 2022 *)
%o A284859 (PARI) T(n, m) = sum(k=0, m, binomial(m, k) * (-1)^(k - m) * (1 + 3*k)^n/m!);
%o A284859 a(n) = sum(m=0, n, T(n, m)); \\ _Indranil Ghosh_, Apr 10 2017
%o A284859 (Python)
%o A284859 from sympy import binomial, factorial
%o A284859 def T(n, m): return sum([binomial(m, k) * (-1)**(k - m) * (1 + 3*k)**n for k in range(m + 1)])//factorial(m)
%o A284859 def a(n): return sum([T(n, k) for k in range(n + 1)])
%o A284859 print([a(n) for n in range(20)]) # _Indranil Ghosh_, Apr 10 2017
%Y A284859 Cf. A000110, A126390, A282629, A284864.
%K A284859 nonn,easy
%O A284859 0,2
%A A284859 _Wolfdieter Lang_, Apr 05 2017