A027634 - OEIS

%I #21 Mar 02 2022 12:01:42

%S 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 356,420,527,623,750,898,1075,1252,1505,1750,2051,2400,2797,3214,3754,

%U 4294,4939,5665,6477,7344,8398,9481,10731,12121,13653

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

%H Andy Huchala, <a href="/A027634/b027634.txt">Table of n, a(n) for n = 0..20000</a>

%H B. Runge, <a href="http://projecteuclid.org/euclid.nmj/1118775400">On Siegel modular forms II</a>, Nagoya Math. J., 138 (1995), 179-197.

%H 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 <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>

%H <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 (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 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 (Sage)

%o R.<x> = PowerSeriesRing(ZZ,80);

%o 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 f = g + x^112*g(1/x);

%o 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 [h.list()[2*i] for i in range(40)] # _Andy Huchala_, Mar 02 2022

%Y Cf. A027633.

%K nonn,easy,nice

%O 0,6

%A _N. J. A. Sloane_