cp's OEIS Frontend

A103620 Number of permutations of n elements admitting a fourth root.

%I A103620 #15 Sep 13 2014 07:37:45
%S A103620 1,1,1,3,9,45,225,1575,11130,100170,897750,9875250,108523800,
%T A103620 1410809400,18332414100,274986211500,4127136413400,70161319027800,
%U A103620 1192076391706200,22649451442417800,430247983427262000,9035207651972502000
%N A103620 Number of permutations of n elements admitting a fourth root.
%H A103620 Alois P. Heinz, <a href="/A103620/b103620.txt">Table of n, a(n) for n = 0..250</a>
%H A103620 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 A103620 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 A103620 with(combinat): with(numtheory): with(padic):
%p A103620 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A103620       `if`(irem(j, mul(p^ordp(4, p), p=factorset(i)))=0, (i-1)!^j*
%p A103620       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A103620     end:
%p A103620 a:= n-> b(n$2):
%p A103620 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 09 2014
%t A103620 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 A103620 Cf. A003483, A103619, A102687, A215716, A215717, A215718.
%Y A103620 Column k=4 of A247005.
%K A103620 nonn
%O A103620 0,4
%A A103620 _Vladeta Jovovic_, Feb 11 2005