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Search: id:A137341
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| A137341 |
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Cooperative games are frequently formulated in terms of partition functions. In particular, the set of players may be divided into various coalitions forming partitions with dierent coalition structures. This recursive sequence identifies the number of partitions in a n-player game where the position of the individual player counts. |
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2
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| 1, 1, 4, 30, 360, 6240, 146160, 4420080, 166924800, 7673823360, 420850080000, 27086342976000, 2018319704755200, 172142484203289600, 16642276683198566400, 1808459441303074560000
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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David, W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung, A Recursive Sequence for the Number of Positioned Partition, International Journal of Algebra, Vol. 2 (2008), No. 4, pp. 181-185.
W. Lucas, and R. Thrall, N-person games in partition function form, Naval Research Logistics Quarterly X, pp.281-298, 1963.
E. T. Bell, Exponential numbers, American Mathematics Monthly, 41 (1934), pp. 411-419.
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FORMULA
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a(0) = 1 : a(n) = SUM(j=0 to n-1) Combination(n-1,j) * n!/j! * a(j)
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EXAMPLE
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a(0) = 1 ;
a(1) = Combination(0,0) * 1/1 * a(0) =1;
a(2) = Combination(1,1) * 2/1 * a(1) + Combination(1,0) * 2/1 * a(0) =4;
a(3) = Combination(2,2) * 6/2 * a(2) + Combination(2,1) *6/1 * a(1) + Combination(2,0) * 6/1 * a(0) =30;
a(4) = Combination(3,3) * 24/6 * a(3) + Combination(3,2) * 24/2 * a(2) + Combination(3,1) *24/1 * a(1) + Combination(3,0 ) * 24/1 * a(0) =360;
a(5) = Combination(4,4)* 120/24 * a(4) + Combination(4,3)* 120/6 * a(3) + Combination(4,2) * 120/2 * a(2) + Combination(4,1) * 120/1 * a(1) + Combination(4,0)* 120/1 * a(0) = 6240.
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PROGRAM
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sage: [factorial(m)*bell_number(m) for m in xrange (0, 17)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 06 2008
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CROSSREFS
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Adjacent sequences: A137338 A137339 A137340 this_sequence A137342 A137343 A137344
Sequence in context: A089918 A132622 A118792 this_sequence A120338 A064050 A119926
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KEYWORD
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nonn
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AUTHOR
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David, W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung (wkyeung(AT)hkbu.edu.hk), Apr 08 2008
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