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A053499 Number of degree-n permutations of order dividing 9.

%I A053499 #26 Sep 08 2022 08:45:00
%S A053499 1,1,1,3,9,21,81,351,1233,46089,434241,2359611,27387801,264333213,
%T A053499 1722161169,16514298711,163094452641,1216239520401,50883607918593,
%U A053499 866931703203699,8473720481213481,166915156382509221,2699805625227141201,28818706120636531023,439756550972215638129,6766483260087819272601,77096822666547068590401,3568144263578808757678251
%N A053499 Number of degree-n permutations of order dividing 9.
%C A053499 Differs from A218003 first at n=27. - _Alois P. Heinz_, Jan 25 2014
%D A053499 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H A053499 Alois P. Heinz, <a href="/A053499/b053499.txt">Table of n, a(n) for n = 0..200</a>
%H A053499 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 A053499 E.g.f.: exp(x + x^3/3 + x^9/9).
%p A053499 a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A053499        add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 3, 9])))
%p A053499     end:
%p A053499 seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 14 2013
%t A053499 CoefficientList[Series[Exp[x+x^3/3+x^9/9], {x, 0, 30}], x]*Range[0, 30]! (* _Jean-François Alcover_, Mar 24 2014 *)
%o A053499 (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^3/3 + x^9/9) )) \\ _G. C. Greubel_, May 15 2019
%o A053499 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3 + x^9/9) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 15 2019
%o A053499 (Sage) m = 30; T = taylor(exp(x + x^3/3 + x^9/9), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 15 2019
%Y A053499 Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261429.
%Y A053499 Column k=9 of A008307.
%K A053499 nonn
%O A053499 0,4
%A A053499 _N. J. A. Sloane_, Jan 15 2000