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A204042
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The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.
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4
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1, 1, 2, 12, 120, 1520, 23160, 413952, 8505280, 197631072, 5125527360, 146787894440, 4601174623584, 156693888150384, 5761055539858528, 227438694372072120, 9596077520725211520, 430920897407809702208, 20520683482765477749120, 1032920864149903149579336, 54797532208320308334631840
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OFFSET
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0,3
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COMMENTS
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Note this sequence counts the functions enumerated by A065440 for which the statement is vacuously true.
a(n) is also the number of partial endofunctions on {1,2,...,n} without fixed points.
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LINKS
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FORMULA
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E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A065440.
a(n) = Sum_{j=0..n} (j-1)^j * binomial(n,j). - Alois P. Heinz, Dec 16 2021
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EXAMPLE
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a(2)=2 because there are two functions f:{1,2}->{1,2} in which all the fixed points are isolated: 1->1,2->2 and 1->2,2->1 (which has no fixed points).
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MAPLE
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a:= n-> add((j-1)^j*binomial(n, j), j=0..n):
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MATHEMATICA
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t = Sum[n^(n-1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[Exp[x] Exp[Log[1/(1-t)]-t], {x, 0, 20}], 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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