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%I #27 Sep 08 2022 08:45:00
%S 1,1,2,6,24,96,576,3312,26496,198144,1691136,14973696,193370112,
%T 2034809856,25087186944,313539434496,4421478721536,58307347556352,
%U 915011420737536,13553664911437824,240637745416421376,3965015057937924096
%N Number of degree-n permutations of order dividing 24.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H Alois P. Heinz, <a href="/A053504/b053504.txt">Table of n, a(n) for n = 0..200</a>
%H L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CJM-1955-020-0">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.
%F 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).
%p a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 4, 6, 8, 12, 24])))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 25 2014
%t 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_ *)
%t With[{nn=30},CoefficientList[Series[Exp[Total[x^#/#&/@Divisors[24]]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 05 2016 *)
%o (PARI) N=30; x='x+O('x^N);
%o Vec(serlaplace(exp(sumdiv(24, d, x^d/d)))) \\ _Gheorghe Coserea_, May 11 2017
%o (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
%o (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
%Y Cf. A000085, A001470, A001472, A053495-A053505, A005388.
%Y Column k=24 of A008307.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 15 2000
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