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A027634 Poincaré (or Molien) series for ring of Siegel modular forms of genus 3 (associated with full modular group Gamma_3).

%I A027634 #21 Mar 02 2022 12:01:42
%S A027634 1,0,1,1,1,2,4,3,7,8,11,15,22,24,37,45,58,75,99,115,156,187,232,288,
%T A027634 356,420,527,623,750,898,1075,1252,1505,1750,2051,2400,2797,3214,3754,
%U A027634 4294,4939,5665,6477,7344,8398,9481,10731,12121,13653
%N A027634 Poincaré (or Molien) series for ring of Siegel modular forms of genus 3 (associated with full modular group Gamma_3).
%H A027634 Andy Huchala, <a href="/A027634/b027634.txt">Table of n, a(n) for n = 0..20000</a>
%H A027634 B. Runge, <a href="http://projecteuclid.org/euclid.nmj/1118775400">On Siegel modular forms II</a>, Nagoya Math. J., 138 (1995), 179-197.
%H A027634 S. Tsuyumine, <a href="https://doi.org/10.2307/2374517">On Siegel modular forms of degree three</a>, Amer. J. Math., 108 (1986), 755-862 and <a href="https://doi.org/10.2307/2374522">Addendum</a>, Amer. J. Math., 108 (1986), 1001-1003.
%H A027634 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A027634 <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>
%H A027634 <a href="/index/Rec#order_54">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 2, -1, -1, 1, 0, -1, -1, -1, 2, -1, -2, 2, 1, 0, 0, -1, 3, 0, -3, 2, 0, 0, 0, -2, 3, 0, -3, 1, 0, 0, -1, -2, 2, 1, -2, 1, 1, 1, 0, -1, 1, 1, -2, 0, 0, 0, 0, -1, 1).
%F A027634 (1-x^8) times Molien series in A027633. That is, same numerator, but denominator is (1-x^4)*(1-x^12)^2*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^30).
%e A027634 1+x^4+x^6+x^8+2*x^10+4*x^12+3*x^14+7*x^16+8*x^18+11*x^20+...
%o A027634 (Sage)
%o A027634 R.<x> = PowerSeriesRing(ZZ,80);
%o A027634 g = 1 + x^4 + x^10 + 3*x^16 - x^18 + 3*x^20 + 2*x^22 + 2*x^24 + 3*x^26 + 4*x^28 + 2*x^30 + 7*x^32 + 3*x^34 + 7*x^36 + 5*x^38 + 9*x^40 + 6*x^42 + 10*x^44 + 8*x^46 + 9*x^50 + 7*x^54 - x^2 + 12*x^52 + 10*x^48 + 7*x^56;
%o A027634 f = g + x^112*g(1/x);
%o A027634 h = (1-x^8)*f(x)*(1 + x^2)/((1 - x^4)*(1 - x^8)*(1 - x^12)^2*(1 - x^14)*(1 - x^18)*(1 - x^20)*(1 - x^30));
%o A027634 [h.list()[2*i] for i in range(40)] # _Andy Huchala_, Mar 02 2022
%Y A027634 Cf. A027633.
%K A027634 nonn,easy,nice
%O A027634 0,6
%A A027634 _N. J. A. Sloane_