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

1, 1, 1, 3, 9, 45, 225, 1575, 11130, 100170, 897750, 9875250, 108523800, 1410809400, 18332414100, 274986211500, 4127136413400, 70161319027800, 1192076391706200, 22649451442417800, 430247983427262000, 9035207651972502000

Offset: 0

Author

Vladeta Jovovic, Feb 11 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 149, Eq. 4.8.2.

Crossrefs

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

Column k=4 of A247005.

Programs

Maple

with(combinat): with(numtheory): with(padic):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(irem(j, mul(p^ordp(4, p), p=factorset(i)))=0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 09 2014

Mathematica

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 *)

Formula

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).