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A209289
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Number of functions f:{1,2,...,2n}->{1,2,...,2n} such that every preimage has an even cardinality.
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4
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1, 2, 40, 2256, 250496, 46063360, 12665422848, 4866544707584, 2490379333697536, 1637285952230719488, 1344814260872574402560, 1349528279475362368847872, 1624638302165034485761966080, 2310920106523435237448955723776, 3834278385523271302103123693142016
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OFFSET
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0,2
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COMMENTS
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Note that the empty set has even cardinality.
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LINKS
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FORMULA
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a(n) = (2n)! * [x^(2n)] cosh(x)^(2n).
a(n) ~ c * n^(2*n) * 2^(2*n) * (1-r)^(2*n) / ((2-r)^n * r^n * exp(2*n)), where r = 0.1664434403990353015638385297757806508596082... is the root of the equation (2/r-1)^(1-r) = exp(2), and c = 1.66711311920192939687232294044843869828... = 2/A085984. - Vaclav Kotesovec, Sep 03 2014, updated Mar 18 2024
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EXAMPLE
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a(1) = 2 because there are 2 functions from {1,2} into {1,2} for which the preimage of both elements has even size: 1,1 (where the preimage of 1 is {1,2} and the preimage of 2 is the empty set) and 2,2 (where the preimage of 1 is the empty set and the preimage of 2 is {1,2}).
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MAPLE
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a:= n-> (2*n)! *coeff(series(cosh(x)^(2*n), x, 2*n+1), x, 2*n):
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MATHEMATICA
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nn=32; Select[Table[n!Coefficient[Series[Cosh[x]^n, {x, 0, nn}], x^n], {n, 0, nn}], #>0&]
a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Cosh[x]^m, {x, 0, m}]]]; (* Michael Somos, Jul 02 2017 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n=2*n; n! * polcoeff( cosh(x + x*O(x^n))^n, n))}; /* Michael Somos, Jul 02 2017 */
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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