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A215720
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The number of functions f:{1,2,...,n}->{1,2,...,n}, endofunctions, such that exactly one nonrecurrent element is mapped into each recurrent element.
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1
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1, 0, 2, 6, 60, 560, 7350, 111552, 2009672, 41378976, 963527850, 25009038560, 716437784172, 22453784964624, 764345507271710, 28085186967504240, 1107971902218683280, 46710909213378892352, 2095883952368863510098, 99724281567446320231104, 5015524096516005263567540
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OFFSET
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0,3
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COMMENTS
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x in {1,2,...,n} is a recurrent element if there is some k such that f^k(x) = x where f^k(x) denotes iterated functional composition. In other words, a recurrent element is in a cycle of the functional digraph.
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LINKS
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FORMULA
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E.g.f.: 1/(1 - x*T(x)) where T(x) is the e.g.f. for A000169.
a(n) = n! * Sum_{i=0..floor(n/2)} i*(n-i)^(n-2*i-1)/(n-2*i)! for n>0, a(0) = 1. - Alois P. Heinz, Aug 22 2012
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EXAMPLE
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a(2) = 2 because we have: (1->1,2->1), (1->2,2->2).
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MAPLE
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a:= n-> `if`(n=0, 1, n! *add(i*(n-i)^(n-2*i-1)/(n-2*i)!, i=0..n/2)):
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MATHEMATICA
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nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[1/(1 - x t) , {x, 0, nn}], x]
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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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