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A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

%I A196834 #23 Aug 22 2021 12:23:11
%S A196834 1,6,37,235,1540,10427,73013,529032,3967195,30785747,247126450,
%T A196834 2050937445,17585497797,155666739742,1421428484337,13377704321695,
%U A196834 129659127547372,1293095848212799,13259069937250169,139671750579429512,1510382932875294447,16754464511605466311
%N A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).
%H A196834 Seiichi Manyama, <a href="/A196834/b196834.txt">Table of n, a(n) for n = 0..500</a>
%F A196834 a(n) = Sum_{m=0..n} A193685(n,m).
%F A196834 E.g.f.: exp(exp(x)+5*x-1).
%F A196834 a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 5) / LambertW(n)^(n + 11/2). - _Vaclav Kotesovec_, Jun 10 2020
%F A196834 a(0) = 1; a(n) = 5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - _Ilya Gutkovskiy_, Jul 03 2020
%e A196834 a(2) = 25 + 11 + 1 = 37.
%p A196834 b:= proc(n, m) option remember;
%p A196834      `if`(n=0, 1, m*b(n-1, m)+b(n-1, m+1))
%p A196834     end:
%p A196834 a:= n-> b(n, 5):
%p A196834 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 22 2021
%t A196834 nmax = 20; CoefficientList[Series[E^(E^x + 5*x - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 10 2020 *)
%Y A196834 Cf. A000110, A005493, A005494, A045379, A196835 (alternating row sums).
%K A196834 nonn,easy
%O A196834 0,2
%A A196834 _Wolfdieter Lang_, Oct 07 2011