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A024716
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a(n) = Sum_{1 <= j <= i <= n} S(i,j), where S(i,j) are Stirling numbers of the second kind.
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5
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1, 3, 8, 23, 75, 278, 1155, 5295, 26442, 142417, 820987, 5034584, 32679021, 223578343, 1606536888, 12086679035, 94951548839, 777028354998, 6609770560055, 58333928795427, 533203744952178, 5039919483399501, 49191925338483847, 495150794633289136
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OFFSET
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1,2
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COMMENTS
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Number of nodes in the set partition tree T(n). See Butler and Sasao. - Michel Marcus, Nov 03 2020
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LINKS
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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.
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FORMULA
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If offset is 0, a(n) = Sum_{i=0..n} binomial(n+1, i+1)*Bell(i) [cf. A000110].
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) = (1/e)*Sum_{k>=1} (k^n - 1)/((k - 1)*(k - 1)!). - Joseph Wheat, Jul 16 2024
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MAPLE
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seq(add(add(Stirling2(k, j), j=1..k), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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