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A053497 Number of degree-n permutations of order dividing 7.

%I A053497 #30 Aug 04 2025 02:12:42
%S A053497 1,1,1,1,1,1,1,721,5761,25921,86401,237601,570241,1235521,892045441,
%T A053497 13348249201,106757164801,604924594561,2722120577281,10344007402561,
%U A053497 34479959558401,24928970490633601,546446134633639681,6281586217487489041,50248618811434961281
%N A053497 Number of degree-n permutations of order dividing 7.
%D A053497 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H A053497 Alois P. Heinz, <a href="/A053497/b053497.txt">Table of n, a(n) for n = 0..200</a>
%H A053497 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 A053497 E.g.f.: exp(x + x^7/7).
%F A053497 a(n) = Sum_{k=0..floor(n/7)} n!/(7^k*k!*(n-7*k)!). - _G. C. Greubel_, Mar 07 2021
%p A053497 a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A053497        add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 7])))
%p A053497     end:
%p A053497 seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 14 2013
%t A053497 CoefficientList[Series[Exp[x+x^7/7], {x, 0, 24}], x]*Range[0, 24]! (* _Jean-François Alcover_, Mar 24 2014 *)
%o A053497 (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^7/7) )) \\ _G. C. Greubel_, May 14 2019
%o A053497 (Magma) R<x>:=PowerSeriesRing(Rationals(), 31); Coefficients(R!(Laplace( Exp(x + x^7/7) ))); // _G. C. Greubel_, May 14 2019, Mar 07 2021
%o A053497 (Sage) f=factorial; [sum(f(n)/(7^j*f(j)*f(n-7*j)) for j in (0..n/7)) for n in (0..30)] # _G. C. Greubel_, May 14 2019
%Y A053497 Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), A118934 (m=4), A052501 (m=5), A293588 (m=6), this sequence (m=7).
%Y A053497 Cf. A000085, A001470, A001472, A005388, A053495 - A053505, A261427.
%Y A053497 Column k=7 of A008307.
%K A053497 nonn
%O A053497 0,8
%A A053497 _N. J. A. Sloane_, Jan 15 2000