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A214854 Number of n-permutations that have exactly two square roots. 1
0, 0, 1, 0, 3, 35, 0, 714, 2835, 35307, 236880, 3342350, 28879158, 461911086, 4916519608, 87798024300, 1112716544355, 21957112744083, 322944848419392, 6986165252185782, 116941654550250258, 2754405555107729418, 51688464405692879688 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)