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%I #24 Jul 04 2023 09:00:28
%S 1,1,2,6,18,66,396,2052,12636,91548,625176,4673736,43575192,377205336,
%T 3624289488,38829340656,397695226896,4338579616272,54018173703456,
%U 641634784488288,8208962893594656,113809776294348576,1526808627197721792,21533423236302943296
%N Number of degree-n permutations of order dividing 6.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H Alois P. Heinz, <a href="/A053496/b053496.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^6/6).
%F D-finite with recurrence a(n) -a(n-1) +(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - _R. J. Mathar_, Jul 04 2023
%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, 6])))
%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, 6}}]]]; 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^6/6], {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 14 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^2/2+x^3/3+x^6/6) )) \\ _G. C. Greubel_, May 14 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^3/3 +x^6/6) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 14 2019
%o (Sage) m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 14 2019
%Y Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261317.
%Y Column k=6 of A008307.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 15 2000
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