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A165684
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Dimension of the space of Siegel cusp forms of genus 2 and dimension 2n (associated with full modular group Gamma_2).
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5
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0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 11, 13, 13, 17, 18, 20, 23, 26, 27, 32, 34, 37, 41, 46, 47, 54, 57, 61, 67, 73, 75, 84, 88, 94, 101, 109, 112, 123, 129, 136, 145, 155, 159, 173, 180, 189, 200, 212, 218, 234, 243, 254, 267, 282, 289, 308, 319
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OFFSET
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1,8
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REFERENCES
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M. Eie, Dimensions of spaces of Siegel cusp forms of degree two and three, AMS, 1984, p. 44-45.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).
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FORMULA
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G.f.: -x^5*(x^6-x-1) / ((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). - Colin Barker, Mar 30 2013
a(n) = 1/1080*n^3 + 1/45*n^2 + O(n). (from g.f.) - Ralf Stephan, Jun 20 2014
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EXAMPLE
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a(5)=1 as the space of Siegel cusp forms of genus 2 and weight 10 is one-dimensional.
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MATHEMATICA
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N1[k_] := 2^(-7)*3^(-3)*5^(-1) (2 k^3 + 96 k^2 - 52 k - 3231); N2[k_] := 2^(-5)*3^(-3)*(17 k - 294) /; Mod[k, 12] == 0; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 254) /; Mod[k, 12] == 2; N2[k_] := 2^(-5)*3^(-3)*(17 k - 86) /; Mod[k, 12] == 4; N2[k_] := 2^(-5)*3^(-3)*(-k - 42) /; Mod[k, 12] == 6; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 2) /; Mod[k, 12] == 8; N2[k_] := 2^(-5)*3^(-3)*(-k + 166) /; Mod[k, 12] == 10; N3[k_] := 2^(-7)*3^(-3)*1131 /; Mod[k, 12] == 0; N3[k_] := 2^(-7)*3^(-3)*(-229) /; Mod[k, 12] == 2; N3[k_] := 2^(-7)*3^(-3)*427 /; Mod[k, 12] == 4; N3[k_] := 2^(-7)*3^(-3)*123 /; Mod[k, 12] == 6; N3[k_] := 2^(-7)*3^(-3)*203 /; Mod[k, 12] == 8; N3[k_] := 2^(-7)*3^(-3)*571 /; Mod[k, 12] == 10; N4[k_] := 5^(-1) /; Mod[k, 5] == 0; N4[k_] := -5^(-1) /; Mod[k, 5] == 3; N4[k_] := 0 /; Mod[k, 5] == 1 || Mod[k, 5] == 2 || Mod[k, 5] == 4; DimSk[k_] := N1[k] + N2[k] + N3[k] + N4[k]/; Mod[k, 2]==0; Table[DimSk[2k], {k, 1, 100}]
CoefficientList[Series[-x^4*(x^6 - x - 1)/((1 - x^2)*(1 - x^3)*(1 - x^5)*(1 - x^6)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{0, 1, 1, 0, 0, 1, -1, -2, -1, 1, 0, 0, 1, 1, 0, -1}, {0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9}, 70] (* Harvey P. Dale, Dec 25 2016 *)
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CROSSREFS
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Cf. A029143 (dimension of the full space of Siegel modular forms of genus 2).
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KEYWORD
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nonn,easy
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AUTHOR
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Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009
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EXTENSIONS
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STATUS
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approved
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