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A008649 Molien series of 3 X 3 upper triangular matrices over GF( 3 ).

%I A008649 #28 Sep 08 2022 08:44:36
%S A008649 1,1,1,2,2,2,3,3,3,5,5,5,7,7,7,9,9,9,12,12,12,15,15,15,18,18,18,22,22,
%T A008649 22,26,26,26,30,30,30,35,35,35,40,40,40,45,45,45,51,51,51,57,57,57,63,
%U A008649 63,63,70,70,70,77,77,77,84,84,84,92,92,92,100,100,100
%N A008649 Molien series of 3 X 3 upper triangular matrices over GF( 3 ).
%C A008649 Number of partitions of n into parts 1, 3 or 9. - _Reinhard Zumkeller_, Aug 12 2011
%D A008649 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%H A008649 G. C. Greubel, <a href="/A008649/b008649.txt">Table of n, a(n) for n = 0..1000</a>
%H A008649 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=219">Encyclopedia of Combinatorial Structures 219</a>
%H A008649 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008649 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, -1, 1).
%F A008649 G.f.: 1/((1-x)*(1-x^3)*(1-x^9)).
%F A008649 a(n) = floor((6*(floor(n/3) +1)*(3*floor(n/3) -n +1) +n^2 +13*n +58)/54). - _Tani Akinari_, Jul 12 2013
%p A008649 1/((1-x)*(1-x^3)*(1-x^9)): seq(coeff(series(%,x,n+1),x,n), n=0..70);
%t A008649 CoefficientList[Series[1/((1-x)*(1-x^3)*(1-x^9)), {x,0,70}], x] (* _G. C. Greubel_, Sep 06 2019 *)
%o A008649 (PARI) my(x='x+O('x^70)); Vec(1/((1-x)*(1-x^3)*(1-x^9))) \\ _G. C. Greubel_, Sep 06 2019
%o A008649 (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^9)) )); // _G. C. Greubel_, Sep 06 2019
%o A008649 (Sage)
%o A008649 def A008649_list(prec):
%o A008649     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008649     return P(1/((1-x)*(1-x^3)*(1-x^9))).list()
%o A008649 A008649_list(70) # _G. C. Greubel_, Sep 06 2019
%K A008649 nonn,easy
%O A008649 0,4
%A A008649 _N. J. A. Sloane_