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

1, 0, 1, 1, 1, 2, 4, 3, 7, 8, 11, 15, 22, 24, 37, 45, 58, 75, 99, 115, 156, 187, 232, 288, 356, 420, 527, 623, 750, 898, 1075, 1252, 1505, 1750, 2051, 2400, 2797, 3214, 3754, 4294, 4939, 5665, 6477, 7344, 8398, 9481, 10731, 12121, 13653

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

			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+...
		

Links

• Andy Huchala, Table of n, a(n) for n = 0..20000
• B. Runge, On Siegel modular forms II, Nagoya Math. J., 138 (1995), 179-197.
• S. Tsuyumine, On Siegel modular forms of degree three, Amer. J. Math., 108 (1986), 755-862 and Addendum, Amer. J. Math., 108 (1986), 1001-1003.
• Index entries for Molien series
• Index entries for sequences related to modular groups
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A027633.

Programs

Sage

R. = PowerSeriesRing(ZZ,80);
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;
f = g + x^112*g(1/x);
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));
[h.list()[2*i] for i in range(40)] # Andy Huchala, Mar 02 2022

Formula

(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).