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A083024 Molien series for action of SL(3,C) on ternary forms of degree 4.

%I A083024 #21 Feb 05 2025 16:19:12
%S A083024 1,1,2,4,7,11,19,29,44,67,98,139,199,275,375,509,678,890,1165,1501,
%T A083024 1916,2431,3053,3801,4711,5788,7063,8580,10353,12420,14841,17633,
%U A083024 20850,24565,28807,33641,39161,45404,52455,60427,69372,79392,90627,103143,117065,132561
%N A083024 Molien series for action of SL(3,C) on ternary forms of degree 4.
%C A083024 These are the coefficients of the expansion in powers of z^4, the other coefficients being zero.
%D A083024 J-M. Kantor, Où en sont les mathématiques. La formule de Molien-Weyl, SMF, Vuibert, p. 79
%H A083024 T. Shioda, <a href="http://www.jstor.org/stable/2373415">On the graded ring of invariants of binary octavics</a>, Amer. J. Math. 89, 1022-1046, 1967.
%H A083024 <a href="/index/Mo#Molien">Index entries for Molien series</a>.
%H A083024 <a href="/index/Rec#order_30">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,0,-2,0,2,0,0,1,0,-1,0,1,0,-1,0,0,-2,0,2,0,1,0,0,-1,-1,1).
%F A083024 G.f.: (1 + z^9 + z^12 + z^15 + 2*z^18 + 3*z^21 + 2*z^24 + 3*z^27 + 4*z^30 + 3*z^33 + 4*z^36 + 4*z^39 + 3*z^42 + 4*z^45 + 3*z^48 + 2*z^51 + 3*z^54 + 2*z^57 + z^60 + z^63 + z^75)/(1-z^3)/(1-z^6)/(1-z^9)/(1-z^12)/(1-z^15)/(1-z^18)/(1-z^27).
%p A083024 seq(coeff(series( (1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 3*x^11 + 4*x^12 + 4*x^13 + 3*x^14 + 4*x^15 + 3*x^16 + 2*x^17 + 3*x^18 + 2*x^19 + x^20 + x^21 + x^25)/(1 - x^1 - x^2 + x^5 + 2*x^7 - 2*x^9 - x^12 + x^14 - x^16 + x^18 + 2*x^21 - 2*x^23 - x^25 + x^28 + x^29 - x^30), x, n+1), x, n), n = 0..45); # _Georg Fischer_, Jan 24 2021
%o A083024 (PARI) a(n)=polcoeff((1+z^9+z^12+z^15+2*z^18+3*z^21+2*z^24+3*z^27+4*z^30+3*z^33 +4*z^36+4*z^39+3*z^42+4*z^45+3*z^48+2*z^51+3*z^54+2*z^57+z^60+z^63+z^75) /(1-z^3)/(1-z^6)/(1-z^9)/(1-z^12)/(1-z^15)/(1-z^18)/(1- z^27)+O(z^(n+1)),n)
%Y A083024 Cf. A008615.
%K A083024 nonn,easy
%O A083024 0,3
%A A083024 _Benoit Cloitre_, Jun 01 2003