A059593 Number of degree-n permutations of order exactly 5.

0, 0, 0, 0, 0, 24, 144, 504, 1344, 3024, 78624, 809424, 4809024, 20787624, 72696624, 1961583624, 28478346624, 238536558624, 1425925698624, 6764765838624, 189239120970624, 3500701266525624, 37764092547420624, 288099608198025624

Offset: 0

Author

Henry Bottomley, Jan 26 2001

Keywords

• nonn

Comments

The number of degree-n permutations of order exactly p (where p is prime) satisfies a(n) =a(n-1) + (1+a(n-p))*(n-1)!/(n-p)! with a(n)=0 if p>n. Also a(n) = Sum_{j=1 to floor[n/p]} n!/(j!*(n-p*j)!*(p^j)).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A001471, A052501.

Column k=5 of A057731. - Alois P. Heinz, Feb 16 2013

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^5/5) )); [Factorial(n-1)*b[n]-1: n in [1..m]]; // G. C. Greubel, May 14 2019

Maple

a:= proc(n) option remember;
      `if`(n<5, 0, a(n-1)+(1+a(n-5))*(n-1)!/(n-5)!)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 25 2014

Mathematica

Table[Sum[n!/(j!*(n-5*j)!*5^j), {j,1,Floor[n/5]}], {n,0,25}] (* G. C. Greubel, May 14 2019 *)

PARI

{a(n) = sum(j=1,floor(n/5), n!/(j!*(n-5*j)!*5^j))}; \\ G. C. Greubel, May 14 2019

Sage

m = 30; T = taylor(exp(x + x^5/5) -exp(x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

Formula

a(n) = a(n - 1) + (1 + a(n - 5))*(n - 1)(n - 2)(n - 3)(n - 4).

a(n) = Sum_{j=1..floor(n/5)} n!/(j!*(n - 5*j)!*(5^j)).

From G. C. Greubel, May 14 2019: (Start)

a(n) = A052501(n) - 1.

E.g.f.: exp(x + x^5/5) - exp(x). (End)