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

0,3

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.

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).

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 *)

Prog

(PARI) N=30; x='x+O('x^N);
Vec(serlaplace(exp(sumdiv(24, d, x^d/d)))) \\ Gheorghe Coserea, May 11 2017
(Magma) m:=30; R<x>:=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
(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

Crossrefs

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

Column k=24 of A008307.

Sequence in context: A152324 A147887 A053502 * A215716 A060725 A150299

Adjacent sequences: A053501 A053502 A053503 * A053505 A053506 A053507

Keyword

nonn

Author

N. J. A. Sloane, Jan 15 2000

Status

approved