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A059593 Number of degree-n permutations of order exactly 5. 4
0, 0, 0, 0, 0, 24, 144, 504, 1344, 3024, 78624, 809424, 4809024, 20787624, 72696624, 1961583624, 28478346624, 238536558624, 1425925698624, 6764765838624, 189239120970624, 3500701266525624, 37764092547420624, 288099608198025624 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 28 13:18 EST 2020. Contains 331321 sequences. (Running on oeis4.)