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

0,6

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

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)

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 *)

Prog

(PARI) {a(n) = sum(j=1, floor(n/5), n!/(j!*(n-5*j)!*5^j))}; \\ G. C. Greubel, May 14 2019
(Magma) m:=30; R<x>:=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
(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

Crossrefs

Cf. A001471, A052501.

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

Sequence in context: A076835 A007900 A158874 * A200194 A182075 A054118

Adjacent sequences: A059590 A059591 A059592 * A059594 A059595 A059596

Keyword

nonn

Author

Henry Bottomley, Jan 26 2001

Status

approved