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A284864 Row sums of Sheffer triangle S2[3,2] given by A225466.

%I A284864 #17 Jun 27 2022 03:26:30
%S A284864 1,5,34,287,2839,31898,399079,5480609,81724300,1311990425,22521232003,
%T A284864 411039834356,7938680371957,161596770440945,3454818798460858,
%U A284864 77340712165173755,1808096791948934755,44038966942707463946,1115155787752634260483,29301563957596710001709
%N A284864 Row sums of Sheffer triangle S2[3,2] given by A225466.
%C A284864 This is a generalization of the Bell sequence A000110 because S2[3,2] is a generalization of the Stirling2 triangle A048993.
%C A284864 For the alternating row sums see A284865.
%F A284864 a(n) = Sum_{k=0..n} A225466(n, k), n >= 0.
%F A284864 E.g.f.: exp(2*x)*exp((exp(3*x)-1)) (Sheffer property).
%F A284864 a(n) = (1/e)*Sum_{m>=0} (1/m!)*(2 + 3*m)^n, n >= 0, (Dobiński type formula).
%F A284864 a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - _Ilya Gutkovskiy_, Jun 21 2022
%F A284864 a(n) ~ 3^n * n^(n + 2/3) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 2/3)). - _Vaclav Kotesovec_, Jun 27 2022
%t A284864 T[n_, k_]:=Sum[Binomial[k, j](-1)^(j - k) (2 + 3j)^n/k!, {j, 0, k}]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 20}] (* _Indranil Ghosh_, Apr 10 2017 *)
%o A284864 (PARI) T(n, k) = sum(j=0, k, binomial(k, j)*(-1)^(j - k)*(2 + 3*j)^n/k!);
%o A284864 a(n) = sum(k=0, n, T(n, k)); \\ _Indranil Ghosh_, Apr 10 2017
%o A284864 (Python)
%o A284864 from sympy import binomial, factorial
%o A284864 def T(n, k): return sum([binomial(k, j)*(-1)**(j - k)*(2 + 3*j)**n/factorial(k) for j in range(k + 1)])
%o A284864 def a(n): return sum([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Apr 10 2017
%Y A284864 Cf. A000110, A225466, A284865, A284859 (case [3,1]).
%K A284864 nonn,easy
%O A284864 0,2
%A A284864 _Wolfdieter Lang_, Apr 10 2017