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

1,2

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

a(n) = (1/e)*Sum_{k>=1} (k^n - 1)/((k - 1)*(k - 1)!). - Joseph Wheat, Jul 16 2024

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