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%I #31 Mar 18 2024 06:47:48
%S 1,2,40,2256,250496,46063360,12665422848,4866544707584,
%T 2490379333697536,1637285952230719488,1344814260872574402560,
%U 1349528279475362368847872,1624638302165034485761966080,2310920106523435237448955723776,3834278385523271302103123693142016
%N Number of functions f:{1,2,...,2n}->{1,2,...,2n} such that every preimage has an even cardinality.
%C Note that the empty set has even cardinality.
%H Seiichi Manyama, <a href="/A209289/b209289.txt">Table of n, a(n) for n = 0..210</a> (terms 0..80 from Alois P. Heinz)
%F a(n) = (2n)! * [x^(2n)] cosh(x)^(2n).
%F a(n) = Sum_{i=0..2*n} (n-i)^(2*n)*binomial(2*n,i). - _Vladimir Kruchinin_, Feb 07 2013
%F 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
%e 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}).
%p a:= n-> (2*n)! *coeff(series(cosh(x)^(2*n), x, 2*n+1), x, 2*n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jan 19 2013
%t nn=32;Select[Table[n!Coefficient[Series[Cosh[x]^n,{x,0,nn}],x^n],{n,0,nn}],#>0&]
%t a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Cosh[x]^m, {x, 0, m}]]]; (* _Michael Somos_, Jul 02 2017 *)
%o (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 */
%Y Cf. A085984.
%K nonn
%O 0,2
%A _Geoffrey Critzer_, Jan 16 2013
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