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Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).
7

%I #23 Aug 22 2021 12:23:11

%S 1,6,37,235,1540,10427,73013,529032,3967195,30785747,247126450,

%T 2050937445,17585497797,155666739742,1421428484337,13377704321695,

%U 129659127547372,1293095848212799,13259069937250169,139671750579429512,1510382932875294447,16754464511605466311

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

%H Seiichi Manyama, <a href="/A196834/b196834.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{m=0..n} A193685(n,m).

%F E.g.f.: exp(exp(x)+5*x-1).

%F a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 5) / LambertW(n)^(n + 11/2). - _Vaclav Kotesovec_, Jun 10 2020

%F 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 a(2) = 25 + 11 + 1 = 37.

%p b:= proc(n, m) option remember;

%p `if`(n=0, 1, m*b(n-1, m)+b(n-1, m+1))

%p end:

%p a:= n-> b(n, 5):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 22 2021

%t nmax = 20; CoefficientList[Series[E^(E^x + 5*x - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 10 2020 *)

%Y Cf. A000110, A005493, A005494, A045379, A196835 (alternating row sums).

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Oct 07 2011