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%I #11 Mar 11 2013 12:36:27
%S 0,0,1,0,3,35,0,714,2835,35307,236880,3342350,28879158,461911086,
%T 4916519608,87798024300,1112716544355,21957112744083,322944848419392,
%U 6986165252185782,116941654550250258,2754405555107729418,51688464405692879688
%N Number of n-permutations that have exactly two square roots.
%C These permutations are of two types: They are composed of exactly one pair of equal even size cycles with at most one fixed point and any number of odd (>=3) size cycles; OR they are any number of odd (>=3) size cycles with exactly two fixed points.
%F E.g.f.: (A(x)*(1+x)+x^2/2)*((1+x)/(1-x))^(1/2)*exp(-x) where A(x) = Sum_{n=2,4,6,8,...} Binomial(2n,n)/2 * x^(2n)/(2n)!
%e a(5) = 35 because we have 20 5-permutations of the type (1,2,3)(4)(5) and 15 of the type (1,2)(3,4)(5). These have 2 square roots:(1,3,2)(4)(5),(1,3,2)(4,5) and (1,3,2,4)(5),(3,1,4,2)(5) respectively.
%t nn=22; a=Sum[Binomial[2n,n]/2x^(2n)/(2n)!, {n,2,nn,2}]; Range[0,nn]! CoefficientList[Series[(a(1+x)+x^2/2) ((1+x)/(1-x))^(1/2) Exp[-x], {x,0,nn}], x]
%Y Cf. A214849, A214851, A003483.
%K nonn
%O 0,5
%A _Geoffrey Critzer_, Mar 08 2013
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