OFFSET
1,2
COMMENTS
Same as A005493, apart from offset. - R. J. Mathar, Sep 23 2011
The strategic behavior of players depends crucially on the coalition structures of a game.
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1995.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..575
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Gábor Czédli, Lattices with lots of congruence energy, arXiv:2205.02294 [math.RA], 2022.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
David W. K. Yeung, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10(2008), 129-136.
FORMULA
a(1) = combination(1,0) = 1, a(2) = combination(2,1) + combination(2,0)= 3, a(n) = {SUM(i=2 to n-1) combination(n,i)} * {SUM(j=1 to i-1) a(n)} + SUM(i=0 to n-1) combination(n,i), for n > 2.
G.f.: 1/U(0) where U(k)= 1 - x*(k+3) - x^2*(k+1)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: 1/(U(0)-x) where U(k)= 1 - x - x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: -G(0)/x^2 where G(k) = 1 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 08 2013
G.f.: Q(0)/x^2 -1/x^2, where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
a(n) = Bell(n+1)-Bell(n) = Sum_{k=1..n} k*Stirling2(n,k). - Alois P. Heinz, May 11 2017
E.g.f.: (exp(x) - 1) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jan 26 2020
EXAMPLE
a(1) = combination(1,0) = 1,
a(2) = combination(2,1) + combination(2,0)= 3,
a(3) = combination(3,2)* a(1) + combination(3,2) + combination(3,1) + combination(3,0)= 10,
a(4) = combination(4,3)* {a(1) + a(2)} + combination(4,2)* a(1) + combination7(4,3)combination(4,2) + combination(4,1) + combination(4,0)= 37,
a(5) = combination(5,4)* {a(1) + a(2) + a(3)} combination(5,3)* {a(1) + a(2)} + combination(5,2)* a(1) + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)= 151.
MATHEMATICA
a[n_] := Sum[Binomial[n, j] BellB[j], {j, 0, n-1}];
Array[a, 24] (* Jean-François Alcover, Aug 19 2018 *)
PROG
(Magma) [&+[k*StirlingSecond(n, k): k in [1..n]]: n in [1..25]]; // Vincenzo Librandi, May 18 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008
STATUS
approved