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%I #31 Sep 08 2022 08:45:00
%S 1,1,1,1,1,1,1,1,1,1,1,3628801,43545601,283046401,1320883201,
%T 4953312001,15850598401,44910028801,115482931201,274271961601,
%U 609493248001,1279935820801,4644633666390681601,106826520356358566401,1281918194457262387201
%N Number of degree-n permutations of order dividing 11.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H Alois P. Heinz, <a href="/A053501/b053501.txt">Table of n, a(n) for n = 0..200</a>
%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2011-2013.
%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^11/11).
%F a(n) = n!*Sum_{k=1..n} (if mod(11*k-n,10)=0 then C(k,(11*k-n)/10)*(11)^((k-n)/10)/k!, else 0), n>0. - _Vladimir Kruchinin_, Sep 10 2010
%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, 11])))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 14 2013
%t a[n_]:= n!*Sum[If[Mod[11*k-n, 10] == 0, Binomial[k, (11*k-n)/10]*11^((k-n)/10)/k!, 0], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Mar 20 2014, after _Vladimir Kruchinin_ *)
%t With[{m = 30}, CoefficientList[Series[Exp[x +x^11/11], {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 15 2019 *)
%o (Maxima) a(n):=n!*sum(if mod(11*k-n,10)=0 then binomial(k,(11*k-n)/10)*(11)^((k-n)/10)/k! else 0,k,1,n); /* _Vladimir Kruchinin_, Sep 10 2010 */
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^11/11) )) \\ _G. C. Greubel_, May 15 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^11/11) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 15 2019
%o (Sage) m = 30; T = taylor(exp(x +x^11/11), 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=11 of A008307.
%K nonn
%O 0,12
%A _N. J. A. Sloane_, Jan 15 2000
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