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%I #43 Sep 08 2022 08:44:36
%S 1,2,3,4,5,6,7,8,9,11,13,15,17,19,21,23,25,27,30,33,36,39,42,45,48,51,
%T 54,58,62,66,70,74,78,82,86,90,95,100,105,110,115,120,125,130,135,141,
%U 147,153,159,165,171,177,183,189,196,203,210,217,224,231,238,245,252
%N Molien series for 3-dimensional group [2,n] = *22n.
%C Number of partitions of n into two kinds of 1's and one kind of 9. - _Joerg Arndt_, Dec 27 2014
%H Vincenzo Librandi, <a href="/A008727/b008727.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=192">Encyclopedia of Combinatorial Structures 192</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,1,-2,1).
%F G.f.: 1/((1-x)^2*(1-x^9)).
%F From _Mitch Harris_, Sep 08 2008: (Start)
%F a(n) = Sum_{j=0..n+9} floor(j/9).
%F a(n-9) = (1/2)*floor(n/9)*(2*n - 7 - 9*floor(n/9)). (End)
%p seq(coeff(series(1/((1-x)^2*(1-x^9)), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Sep 09 2019
%t Drop[Accumulate[Floor[Range[70]/9]], 8] (* _Jean-François Alcover_, Mar 27 2013 *)
%t CoefficientList[Series[1/(1-x)^2/(1-x^9), {x,0,70}], x] (* _Vincenzo Librandi_, Jun 11 2013 *)
%t LinearRecurrence[{2,-1,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,11,13},120] (* _Harvey P. Dale_, Feb 13 2022 *)
%o (PARI) Vec(1/(1-x)^2/(1-x^9)+O(x^66)) /* _Joerg Arndt_, Mar 27 2013 */
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^9)) )); // _G. C. Greubel_, Sep 09 2019
%o (Sage)
%o def A008727_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(1/((1-x)^2*(1-x^9))).list()
%o A008727_list(70) # _G. C. Greubel_, Sep 09 2019
%o (GAP) a:=[1,2,3,4,5,6,7,8,9,11,13];; for n in [12..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # _G. C. Greubel_, Sep 09 2019
%Y Cf. A001840, A001972, A008724, A008725, A008726, A008732. - _Vladimir Joseph Stephan Orlovsky_, Mar 14 2010
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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