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%I #16 Sep 08 2022 08:44:49
%S 1,15,135,870,3992,14142,41400,104870,237288,490654,942888,1705510,
%T 2932344,4829246,7664856,11782374,17612360,25686558,36652744,51290598,
%U 70528600,95461950,127371512,167743782,218291880,280977566,358034280,451991206,565698360
%N Molien series for Gamma_3(2).
%H Colin Barker, <a href="/A027630/b027630.txt">Table of n, a(n) for n = 0..1000</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/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: (1 - x^4)*(1 + 7*x + 43*x^2 + 154*x^3 + 43*x^4 + 7*x^5 + x^6) / (1-x)^8.
%F From _Colin Barker_, Jan 03 2017: (Start)
%F a(n) = 2*(3060 -6138*n +5933*n^2 -3120*n^3 +1100*n^4 -192*n^5 +32*n^6)/45 for n>2.
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9. (End)
%F E.g.f.: (-12150 -1350*x -45*x^2 + (12240 -9540*x +9540*x^2 +6240*x^3 + 5040*x^4 +1152*x^5 +128*x^6)*exp(x))/90. - _G. C. Greubel_, Feb 01 2020
%p 1, seq( `if`(n<3, 15*(2*n-1)^2, 2*(3060 -6138*n +5933*n^2 -3120*n^3 +1100*n^4 -192*n^5 +32*n^6)/45), n=1..30); # _G. C. Greubel_, Feb 01 2020
%t Table[If[n==0, 1, If[n<3, 15*(2*n-1)^2, 2*(3060 -6138*n +5933*n^2 -3120*n^3 + 1100*n^4 -192*n^5 +32*n^6)/45]], {n,0,30}] (* _G. C. Greubel_, Feb 01 2020 *)
%o (PARI) Vec((1+x)*(1+x^2)*(1+7*x+43*x^2+154*x^3+43*x^4+7*x^5+x^6) / (1-x)^7 + O(x^30)) \\ _Colin Barker_, Jan 03 2017
%o (Magma) [1,15,135] cat [2*(3060 -6138*n +5933*n^2 -3120*n^3 +1100*n^4 -192*n^5 +32*n^6)/45: n in [3..30]]; // _G. C. Greubel_, Feb 01 2020
%o (Sage) [1,15,135]+[2*(3060 -6138*n +5933*n^2 -3120*n^3 +1100*n^4 -192*n^5 +32*n^6)/45 for n in (3..30)] # _G. C. Greubel_, Feb 01 2020
%o (GAP) Concatenation([1,15,135], List([3..30], n-> 2*(3060 -6138*n +5933*n^2 -3120*n^3 +1100*n^4 -192*n^5 +32*n^6)/45 )); # _G. C. Greubel_, Feb 01 2020
%Y Cf. A027628, A027629.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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