A053504 Number of degree-n permutations of order dividing 24.

1, 1, 2, 6, 24, 96, 576, 3312, 26496, 198144, 1691136, 14973696, 193370112, 2034809856, 25087186944, 313539434496, 4421478721536, 58307347556352, 915011420737536, 13553664911437824, 240637745416421376, 3965015057937924096

Offset: 0

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=24 of A008307.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^3/3 +x^4/4 +x^6/6 +x^8/8 +x^12/12 +x^24/24) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

Maple

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 4, 6, 8, 12, 24])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 25 2014

Mathematica

a[n_]:= a[n] = If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 4, 6, 8, 12, 24}}]]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
With[{nn=30},CoefficientList[Series[Exp[Total[x^#/#&/@Divisors[24]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 05 2016 *)

PARI

N=30; x='x+O('x^N);
Vec(serlaplace(exp(sumdiv(24, d, x^d/d)))) \\ Gheorghe Coserea, May 11 2017

Sage

m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^4/4 +x^6/6 +x^8/8 +x^12/12 +x^24/24), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

Formula

E.g.f.: exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^8/8 + x^12/12 + x^24/24).