A029143 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.

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, 39, 42, 46, 52, 52, 60, 63, 67, 73, 80, 81, 91, 95, 101, 108, 117, 119, 131, 137, 144, 153, 164, 167, 182, 189, 198, 209, 222

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

Number of partitions of n into parts 2, 3, 5, and 6. - Joerg Arndt, Jun 21 2014

References

• H. Klingen, Intro. lectures on Siegel modular forms, Cambridge, p. 123, Corollary.
• L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 31).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
• W. C. Huffman, The biweight enumerator of self-orthogonal binary codes, Discr. Math. Vol. 26 1979, pp. 129-143.
• J. Igusa, On Siegel modular forms of genus 2, Amer. J. Math., 84 (1962), 175-200.
• Index entries for Molien series
• 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).

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)
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 *)

Formula

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

a(n) ~ 1/1080*n^3. - Ralf Stephan, Apr 29 2014

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

Extensions

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