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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

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.

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

Adjacent sequences:  A024713 A024714 A024715 * A024717 A024718 A024719

KEYWORD

nonn,easy,nice

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified October 27 12:08 EDT 2021. Contains 348276 sequences. (Running on oeis4.)