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A024716 a(n) = Sum_{1 <= j <= i <= n} S(i,j), where S(i,j) are Stirling numbers of the second kind. 5
1, 3, 8, 23, 75, 278, 1155, 5295, 26442, 142417, 820987, 5034584, 32679021, 223578343, 1606536888, 12086679035, 94951548839, 777028354998, 6609770560055, 58333928795427, 533203744952178, 5039919483399501, 49191925338483847, 495150794633289136 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row sums of triangle A137649. - Gary W. Adamson, Feb 01 2008
Number of nodes in the set partition tree T(n). See Butler and Sasao. - Michel Marcus, Nov 03 2020
LINKS
Jon T. Butler and Tsutomu Sasao, A set partition number system, Australasian Journal of Combinatorics, Volume 65(2) (2016), 152-169. See Table 4, p. 167.
FORMULA
If offset is 0, a(n) = Sum_{i=0..n} binomial(n+1, i+1)*Bell(i) [cf. A000110].
Partial sums of Bell numbers. - Vladeta Jovovic, Mar 16 2003
From Sergei N. Gladkovskii, Dec 20 2012 and Jan 2013: (Start)
Recursively defined continued fractions:
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))).
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))).
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)).
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))).
G.f.: 1/(G(0) - x)/(1 - x), where G(k) = 1 - x*(k + 1)/(1 - x/G(k+1)). (End)
a(n) ~ Bell(n) / (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
MAPLE
seq(add(add(Stirling2(k, j), j=1..k), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007
MATHEMATICA
Accumulate[Table[BellB[n], {n, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
CROSSREFS
Equals A005001(n+1) - 1.
First column of triangle A101908.
Cf. A137649.
Sequence in context: A148778 A099265 A099266 * A189359 A125782 A343176
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 19 06:53 EDT 2024. Contains 370953 sequences. (Running on oeis4.)