%I #45 Jan 31 2022 06:48:26

%S 1,0,1,1,1,2,3,2,4,4,5,6,8,7,10,11,12,14,17,16,21,22,24,27,31,31,37,

%T 39,42,46,52,52,60,63,67,73,80,81,91,95,101,108,117,119,131,137,144,

%U 153,164,167,182,189,198,209,222

%N Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). Molien series for u.g.g.r. #31 of order 46080. Poincaré series [or Poincare series] for ring of even weight Siegel 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 2k (for the full modular group Gamma_2). Also: Number of solutions of 4x+6y+10z+12w=k in nonnegative integers x,y,z,w. - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

%C Number of partitions of n into parts 2, 3, 5, and 6. - _Joerg Arndt_, Jun 21 2014

%D H. Klingen, Intro. lectures on Siegel modular forms, Cambridge, p. 123, Corollary.

%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 31).

%H Seiichi Manyama, <a href="/A029143/b029143.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)

%H W. C. Huffman, <a href="http://dx.doi.org/10.1016/0012-365X(79)90119-5">The biweight enumerator of self-orthogonal binary codes</a>, Discr. Math. Vol. 26 1979, pp. 129-143.

%H J. Igusa, <a href="https://www.jstor.org/stable/2372812">On Siegel modular forms of genus 2</a>, Amer. J. Math., 84 (1962), 175-200.

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <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 a(n) = A165684(n) + A008615(n+2). - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

%F a(n) ~ 1/1080*n^3. - _Ralf Stephan_, Apr 29 2014

%F a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=1, a(5)=2, a(6)=3, a(7)=2, a(8)=4, a(9)=4, a(10)=5, a(11)=6, a(12)=8, a(13)=7, a(14)=10, a(15)=11, a(n)= a(n-2)+ a(n-3)+a(n-6)-a(n-7)- 2*a(n-8)-a(n-9)+a(n-10)+a(n-13)+ a(n-14)- a(n-16). - _Harvey P. Dale_, May 12 2015

%p M := Matrix(16, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 6, 10, 13, 14])) then 1 elif j=1 and member(i, [7, 9, 16]) then -1 elif j=1 and i=8 then -2 else 0 fi): a:= n -> (M^(n))[1,1]: seq(a(n), n=0..54); # _Alois P. Heinz_, Jul 25 2008

%t CoefficientList[Series[1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)),{x,0,54}],x] (* _Jean-François Alcover_, Mar 20 2011 *)

%t LinearRecurrence[{0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1},{1,0,1,1,1,2,3,2,4,4,5,6,8,7,10,11},60] (* _Harvey P. Dale_, May 12 2015 *)

%Y Cf. A027640 for the dimension of even and odd weight Siegel modular forms. See A165684 (resp. A165685) for the corresponding space of cusp forms. - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

%K nonn,easy,nice

%O 0,6

%A _N. J. A. Sloane_

%E Definition corrected by Kilian Kilger (kilian(AT)nihilnovi.de), Sep 25 2009