A214849 - OEIS

A214849

Number of n-permutations having all cycles of odd length and at most one fixed point.

%I #19 Oct 09 2013 08:31:42

%S 1,1,0,2,8,24,184,1000,8448,66752,670976,6771456,80540800,981684352,

%T 13555365888,193136762624,3042586824704,49558509465600,

%U 877951349198848,16081833643651072,316609129672114176,6439690754082062336,139521103623589068800

%N Number of n-permutations having all cycles of odd length and at most one fixed point.

%C a(n) is also the number of n-permutations with exactly one square root. Cf. A003483 which counts n-permutations with at least one square root.

%H Vincenzo Librandi, <a href="/A214849/b214849.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: (1 + x)*((1+x)/(1-x))^(1/2)*exp(-x).

%F a(n) ~ 4*n^n/exp(n+1). - _Vaclav Kotesovec_, Oct 08 2013

%e a(6)= 184 because we have 144 6-permutations of the type (1,2,3,4,5)(6) and 40 of the type (1,2,3)(4,5,6). These have exactly one square root: (1,4,2,5,3)(6) and (1,3,2)(4,6,5).

%t nn=22; Range[0,nn]! CoefficientList[Series[(1+x)((1+x)/(1-x))^(1/2) Exp[-x], {x,0,nn}], x]

%K nonn

%O 0,4

%A _Geoffrey Critzer_, Mar 08 2013