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%I A165684 #27 Mar 19 2024 10:25:17
%S A165684 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,
%T A165684 41,46,47,54,57,61,67,73,75,84,88,94,101,109,112,123,129,136,145,155,
%U A165684 159,173,180,189,200,212,218,234,243,254,267,282,289,308,319
%N A165684 Dimension of the space of Siegel cusp forms of genus 2 and dimension 2n (associated with full modular group Gamma_2).
%D A165684 M. Eie, Dimensions of spaces of Siegel cusp forms of degree two and three, AMS, 1984, p. 44-45.
%H A165684 Harvey P. Dale, <a href="/A165684/b165684.txt">Table of n, a(n) for n = 1..1000</a>
%H A165684 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).
%F A165684 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
%F A165684 a(n) = 1/1080*n^3 + 1/45*n^2 + O(n). (from g.f.) - _Ralf Stephan_, Jun 20 2014
%e A165684 a(5)=1 as the space of Siegel cusp forms of genus 2 and weight 10 is one-dimensional.
%t A165684 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}]
%t A165684 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 *)
%t A165684 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 *)
%Y A165684 Cf. A029143 (dimension of the full space of Siegel modular forms of genus 2).
%K A165684 nonn,easy
%O A165684 1,8
%A A165684 Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009
%E A165684 More terms from _Wesley Ivan Hurt_, Jun 20 2014