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Number of embedded coalitions in an n-person game where the position of the individual player is important.
1

%I #4 Sep 18 2011 16:19:46

%S 1,6,60,888,18120,485280,16445520,685722240,34411184640,2041544736000,

%T 141106965753600,11223409849344000,1016591564596608000,

%U 103921686070339737600,11896153817325313536000

%N Number of embedded coalitions in an n-person game where the position of the individual player is important.

%C See A005493 for references and links.

%F a(1) = 1 * SUM(i=0 to 0)combination(1,i) = 1, a(2) = 2 * SUM(i=0 to 1)combination(2,i) = 6, a(n) = n![SUM(i=2 to n-1) combination(n,i) * {SUM(j=1 to i-1) * a(j)/j! } + SUM(i=0 to n-1) combination(n,i)], for n > 2.

%e a(1) = 1 * SUM(i=0 to 0)combination(1,i) = 1,

%e a(2) = 2 * SUM(i=0 to 1)combination(2,i) = 6,

%e a(3) = 3![combination(3,2) * a(1)/1! + combination(3,2) + combination(3,1) + combination(3,0)] = 60,

%e a(4) = 4![combination(4,3) * {a(2)/2! + a(1)/1!} + combination(4,2) * a(1)/1! + combination(4,3) + combination(4,2) + combination(4,1) + combination(4,0)] = 888,

%e a(5) = 5![combination(5,4) * {a(3)/3! + a(2)/2! + a(1)/1!} + combination(5,3) * {a(2)/2! + a(1)/1!} + combination(5,2) * a(1)/1! + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)] = 18120.

%K nonn

%O 1,2

%A David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008