%I #26 Mar 02 2022 12:02:16
%S 1,0,1,1,2,2,5,4,9,10,16,19,31,34,53,64,89,109,152,179,245,296,384,
%T 467,601,716,911,1090,1351,1614,1986,2342,2856,3364,4037,4742,5653,
%U 6578,7791,9036,10592,12243,14268,16380,18990,21724,24999
%N Molien series for full 8 X 8 Siegel modular group H_3 of order 371589120.
%H Jean-François Alcover, <a href="/A027633/b027633.txt">Table of n, a(n) for n = 0..999</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 <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_58">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 1, -1, 2, -1, -1, 1, -2, 0, 0, -2, 2, 0, -1, 3, -1, 1, 2, -3, 2, 0, -3, 3, -3, 0, 3, -4, 3, 0, -3, 3, -3, 0, 2, -3, 2, 1, -1, 3, -1, 0, 2, -2, 0, 0, -2, 1, -1, -1, 2, -1, 1, 0, 0, 1, -1).
%F Reference gives explicit formula for Molien series.
%F Molien series is 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)),
%F where f(x) = g(x) + x^112*g(1/x), g(x) = 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.
%e 1 + x^4 + x^6 + 2*x^8 + 2*x^10 + 5*x^12 + 4*x^14 + 9*x^16 + 10*x^18 + 16*x^20 + ...
%o (Sage)
%o R.<x> = PowerSeriesRing(ZZ,40);
%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 = 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(20)] # _Andy Huchala_, Mar 02 2022
%Y Cf. A027672, A027638. Bisection gives A039946.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_