A214854 Number of n-permutations that have exactly two square roots.

0, 0, 1, 0, 3, 35, 0, 714, 2835, 35307, 236880, 3342350, 28879158, 461911086, 4916519608, 87798024300, 1112716544355, 21957112744083, 322944848419392, 6986165252185782, 116941654550250258, 2754405555107729418, 51688464405692879688

Offset: 0

Author

Geoffrey Critzer, Mar 08 2013

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A214849, A214851, A003483.

Programs

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]

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)!