%I #27 Aug 05 2022 07:46:27
%S 1,0,0,0,1,0,1,0,1,0,2,0,3,0,2,0,4,0,4,0,5,0,6,0,8,0,7,0,10,0,11,0,12,
%T 0,14,1,17,0,16,1,21,1,22,1,24,2,27,3,31,2,31,4,37,4,39,5,42,6,46,8,
%U 52,7,52,10,60,11,63,12,67,14
%N Poincaré series [or Poincare series] for ring of modular forms of genus 2.
%C a(k) for k>0 is the dimension of the space of Siegel modular forms of genus 2 and weight k (for the full modular group Gamma_2). - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009
%D B. Runge, On Siegel modular forms I, J. Reine Angew. Math., 436 (1993), 57-85.
%H Colin Barker, <a href="/A027640/b027640.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Igusa, <a href="http://www.jstor.org/stable/2373172">On Siegel modular forms of genus 2 (II)</a>, Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
%H <a href="/index/Rec#order_27">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,1,1,0,0,-1,-1,-1,1,0,0,1,-1,-1,-1,0,0,1,1,1,0,0,0,-1).
%F G.f.: (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)).
%t Table[SeriesCoefficient[Series[(1+t^(35))/((1-t^4) (1-t^6)(1-t^(10)) (1-t^(12))), {t, 0,100}], i], {i, 0, 100}] (* Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009 *)
%o (PARI) Vec((1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) + O(x^100)) \\ _Colin Barker_, Jul 27 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) )); // _G. C. Greubel_, Aug 04 2022
%o (Sage)
%o def A027640_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) ).list()
%o A027640_list(100) # _G. C. Greubel_, Aug 04 2022
%Y Cf. A165685 for the corresponding dimension of the space of cusp forms. - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009
%K nonn,easy
%O 0,11
%A _N. J. A. Sloane_