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Number of permutations of n elements admitting a fourth root.
7

%I #15 Sep 13 2014 07:37:45

%S 1,1,1,3,9,45,225,1575,11130,100170,897750,9875250,108523800,

%T 1410809400,18332414100,274986211500,4127136413400,70161319027800,

%U 1192076391706200,22649451442417800,430247983427262000,9035207651972502000

%N Number of permutations of n elements admitting a fourth root.

%H Alois P. Heinz, <a href="/A103620/b103620.txt">Table of n, a(n) for n = 0..250</a>

%H H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 149, Eq. 4.8.2.

%F E.g.f.: ((1+x)/(1-x))^(1/2)*Product(1/2*cos(1/2*x^(2*m)/m)+1/2*cosh(1/2*x^(2*m)/m), m = 1 .. infinity).

%p with(combinat): with(numtheory): with(padic):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p `if`(irem(j, mul(p^ordp(4, p), p=factorset(i)))=0, (i-1)!^j*

%p multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 09 2014

%t CoefficientList[Series[((1+x)/(1-x))^(1/2) * Product[1/2*Cos[1/2*x^(2*m)/m] + 1/2*Cosh[1/2*x^(2*m)/m],{m,1,20}],{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Sep 13 2014 *)

%Y Cf. A003483, A103619, A102687, A215716, A215717, A215718.

%Y Column k=4 of A247005.

%K nonn

%O 0,4

%A _Vladeta Jovovic_, Feb 11 2005