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A027633
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Molien series for full 8 X 8 Siegel modular group H_3 of order 371589120.
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4
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1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 16, 19, 31, 34, 53, 64, 89, 109, 152, 179, 245, 296, 384, 467, 601, 716, 911, 1090, 1351, 1614, 1986, 2342, 2856, 3364, 4037, 4742, 5653, 6578, 7791, 9036, 10592, 12243, 14268, 16380, 18990, 21724, 24999
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OFFSET
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0,5
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LINKS
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Index entries for linear recurrences with constant coefficients, 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).
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FORMULA
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Reference gives explicit formula for Molien series.
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)),
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.
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EXAMPLE
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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 + ...
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PROG
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(Sage)
R.<x> = PowerSeriesRing(ZZ, 40);
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 = 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(20)] # Andy Huchala, Mar 02 2022
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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