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A053498 Number of degree-n permutations of order dividing 8.

%I A053498 #21 Sep 08 2022 08:45:00
%S A053498 1,1,2,4,16,56,256,1072,11264,78976,672256,4653056,49810432,433429504,
%T A053498 4448608256,39221579776,607251736576,7244686764032,101611422797824,
%U A053498 1170362064019456,19281174853615616,261583327556386816,4084459360167657472,54366023748591386624
%N A053498 Number of degree-n permutations of order dividing 8.
%D A053498 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H A053498 Alois P. Heinz, <a href="/A053498/b053498.txt">Table of n, a(n) for n = 0..200</a>
%H A053498 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 A053498 E.g.f.: exp(x + x^2/2 + x^4/4 + x^8/8).
%p A053498 a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A053498        add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 4, 8])))
%p A053498     end:
%p A053498 seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 14 2013
%t A053498 CoefficientList[Series[Exp[x+x^2/2+x^4/4+x^8/8], {x, 0, 23}], x]*Range[0, 23]! (* _Jean-François Alcover_, Mar 24 2014 *)
%o A053498 (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^2/2 +x^4/4 +x^8/8) )) \\ _G. C. Greubel_, May 14 2019
%o A053498 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^4/4 +x^8/8) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 14 2019
%o A053498 (Sage) m = 30; T = taylor(exp(x +x^2/2 +x^4/4 +x^8/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 14 2019
%Y A053498 Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261428.
%Y A053498 Column k=8 of A008307.
%K A053498 nonn
%O A053498 0,3
%A A053498 _N. J. A. Sloane_, Jan 15 2000