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A273442
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Number of endofunctions on [2n] with exactly n cycles.
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2
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1, 3, 95, 5595, 484729, 55545735, 7923937307, 1353285904971, 269240651261153, 61157779792168059, 15617503320899550135, 4429016799173481942427, 1381112305978592892946825, 469689278931628969590283855, 173002815169302537782725771395
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (2*n)!/n! * [x^(2*n)] (-log(1+LambertW(-x)))^n.
a(n) ~ c * d^n * n^(n-1/2), where d = 2^(4-r) * exp(1-r) * (2-r)^(r-2) * log(s) / (1-1/s)^r = 10.40858458700790823344027277763248832..., where r = 1.2672171362228848078038115564503589940694831794020694762759870935... is the root of the equation r*log(s) * (-1 + (r-s)* log((2*(s-1))/(s*(2-r)))) = 1 - s, where s = -r*LambertW(-1, -exp(-1/r)/r) = 1.5782614856055967129193228312616913... and c = 0.336740238865974324583136447665761... - Vaclav Kotesovec, Nov 01 2016, extended Aug 28 2017
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MATHEMATICA
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Table[(2*n)!/n! * SeriesCoefficient[(-Log[1+LambertW[-x]])^n, {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 01 2016 *)
Flatten[{1, Table[Sum[Binomial[2*n-1, k] * (2*n)^(2*n-1-k) * Abs[StirlingS1[k+1, n]], {k, 0, 2*n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 01 2016 *)
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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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