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

0,6

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).

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

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 *)

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

Sequence in context: A364346 A182762 A173997 * A363263 A153846 A284383

Adjacent sequences: A029140 A029141 A029142 * A029144 A029145 A029146

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved