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A038154
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a(n) = n! * Sum_{k=0..n-2} 1/k!.
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8
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0, 0, 2, 12, 60, 320, 1950, 13692, 109592, 986400, 9864090, 108505100, 1302061332, 16926797472, 236975164790, 3554627472060, 56874039553200, 966858672404672, 17403456103284402, 330665665962403980, 6613313319248079980, 138879579704209680000
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OFFSET
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0,3
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COMMENTS
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The number of rank-orderings of (>=2)-element subsets of an n-set. (Counts nontrivial votes in a rank-ordering voting system.) E.g., a(5) = 320 = 120+120+60+20 because of 5-, 4-, 3- and 2-element subsets. - Warren D. Smith, Jul 06 2005
a(n) is the number of simple cycles through a vertex of the complete graph K_(n+1) on n+1 vertices [Hassani]. For example, in the complete graph K_4 with vertex set {A,B,C,D} there are a(3) = 12 simple cycles at the vertex A, namely the six 3-cycles ABCA, ABDA, ACBA, ACDA, ADBA and ADCA and the six 4-cycles ABCDA, ABDCA, ACBDA, ACDBA, ADBCA and ADCBA. The sum of the lengths of the cycles at a vertex of K_n is equal to A141834(n). - Peter Bala, Jul 09 2008
See A000522 for the number of paths between a pair of distinct vertices of K_n. - Peter Bala, Jul 09 2008
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LINKS
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FORMULA
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a(n) = n*a(n-1) + n*(n-1), for n>=3, a(2)=2 and a(3) = 12. - Ian Myers, Jul 19 2012
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EXAMPLE
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0=1*0+0, 2=2*0+2, 12=3*2+6, 60=4*12+12, 320 = 5*60+20, ... - Gary Detlefs, May 20 2010
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MATHEMATICA
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Table[n!Sum[1/k!, {k, 0, n-2}], {n, 0, 30}] (* Harvey P. Dale, Jun 04 2012 *)
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PROG
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(PARI) main(size)=my(k); vector(size, n, (n-1)!*sum(k=0, n-3, 1/k!)); \\ Anders Hellström, Jul 14 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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