A024716 a(n) = Sum_{1 <= j <= i <= n} S(i,j), where S(i,j) are Stirling numbers of the second kind.
1, 3, 8, 23, 75, 278, 1155, 5295, 26442, 142417, 820987, 5034584, 32679021, 223578343, 1606536888, 12086679035, 94951548839, 777028354998, 6609770560055, 58333928795427, 533203744952178, 5039919483399501, 49191925338483847, 495150794633289136
Offset: 1
Links
- T. D. Noe, Table of n, a(n) for n = 1..100
- 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.
Programs
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Maple
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
Accumulate[Table[BellB[n], {n, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
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
a(n) = (1/e)*Sum_{k>=1} (k^n - 1)/((k - 1)*(k - 1)!). - Joseph Wheat, Jul 16 2024
Comments