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

0,5

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.

Links

Table of n, a(n) for n=0..22.

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

Cf. A214849, A214851, A003483.

Sequence in context: A182382 A222787 A180126 * A054287 A176761 A246824

Adjacent sequences: A214851 A214852 A214853 * A214855 A214856 A214857

Keyword

nonn

Author

Geoffrey Critzer, Mar 08 2013

Status

approved