|
%I #48 Jul 28 2021 09:10:05
%S 1,3,8,23,75,278,1155,5295,26442,142417,820987,5034584,32679021,
%T 223578343,1606536888,12086679035,94951548839,777028354998,
%U 6609770560055,58333928795427,533203744952178,5039919483399501,49191925338483847,495150794633289136
%N a(n) = Sum_{1 <= j <= i <= n} S(i,j), where S(i,j) are Stirling numbers of the second kind.
%C Row sums of triangle A137649. - _Gary W. Adamson_, Feb 01 2008
%C Number of nodes in the set partition tree T(n). See Butler and Sasao. - _Michel Marcus_, Nov 03 2020
%H T. D. Noe, <a href="/A024716/b024716.txt">Table of n, a(n) for n = 1..100</a>
%H Jon T. Butler and Tsutomu Sasao, <a href="https://ajc.maths.uq.edu.au/pdf/65/ajc_v65_p152.pdf">A set partition number system</a>, Australasian Journal of Combinatorics, Volume 65(2) (2016), 152-169. See Table 4, p. 167.
%F If offset is 0, a(n) = Sum_{i=0..n} binomial(n+1, i+1)*Bell(i) [cf. A000110].
%F Partial sums of Bell numbers. - _Vladeta Jovovic_, Mar 16 2003
%F From _Sergei N. Gladkovskii_, Dec 20 2012 and Jan 2013: (Start)
%F Recursively defined continued fractions:
%F G.f.: G(0)/(1-x) where G(k) = 1 - 2*x*(k + 1)/((2*k + 1)*(2*x*k + x - 1) - x*(2*k + 1)*(2*k + 3)*(2*x*k + x - 1)/(x*(2*k + 3) - 2*(k + 1)*(2*x*k + 2*x - 1)/G(k+1))).
%F G.f.: (G(0) - 1)/(1 - x) where G(k) = 1 + (1 - x)/(1 - x*(k + 1))/(1 - x/(x + (1 -x)/G(k+1))).
%F G.f.: (S - 1)/(1 - x), where S = (1/(1 - x)) * Sum_{k>=0} ((1 + (1 - x)/(1 - x -x*k))*x^k / Product_{i=1..k-1} (1 - x - x*i)).
%F G.f.: ((G(0) - 2)/(2*x - 1) - 1)/(1 - x)/x where G(k) = 2 - 1/(1 - k*x)/(1 - x/(x - 1/G(k+1))).
%F G.f.: 1/(G(0) - x)/(1 - x), where G(k) = 1 - x*(k + 1)/(1 - x/G(k+1)). (End)
%F a(n) ~ Bell(n) / (1 - LambertW(n)/n). - _Vaclav Kotesovec_, Jul 28 2021
%p seq(add(add(Stirling2(k, j),j=1..k), k=1..n), n=1..23); # _Zerinvary Lajos_, Dec 04 2007
%t Accumulate[Table[BellB[n], {n, 40}]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%Y Equals A005001(n+1) - 1.
%Y First column of triangle A101908.
%Y Cf. A137649.
%K nonn,easy,nice
%O 1,2
%A _Clark Kimberling_
| 