OFFSET
0,5
COMMENTS
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.
FORMULA
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)!
EXAMPLE
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.
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Mar 08 2013
STATUS
approved