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%I #20 Sep 08 2022 08:45:00
%S 1,1,2,6,18,90,540,3060,20700,145980,1459800,13854600,140059800,
%T 1514748600,15869034000,285268878000,4109761962000,59488383690000,
%U 935767530036000,13364309726748000,240338216104020000,4540941256642020000,79739974380153240000
%N Number of degree-n permutations of order dividing 30.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H Alois P. Heinz, <a href="/A053505/b053505.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^5/5 + x^6/6 + x^10/10 + x^15/15 + x^30/30).
%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, 5, 6, 10, 15, 30])))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 14 2013
%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, 5, 6, 10, 15, 30}}]]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 03 2014, after _Alois P. Heinz_ *)
%t With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^5/5 +x^6/6 + x^10/10 +x^15/15 +x^30/30], {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 15 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^2/2 +x^3/3 +x^5/5 + x^6/6 +x^10/10 +x^15/15 +x^30/30) )) \\ _G. C. Greubel_, May 15 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^3/3 +x^5/5 +x^6/6 +x^10/10 +x^15/15 +x^30/30) )); [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^5/5 +x^6/6 +x^10/10 +x^15/15 +x^30/30), 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=30 of A008307.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 15 2000
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