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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

J. Igusa, On Siegel modular forms of genus 2, Amer. J. Math., 84 (1962), 175-200.

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.

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: A179806 A182762 A173997 * A153846 A284383 A072406

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

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Last modified December 15 04:20 EST 2019. Contains 329991 sequences. (Running on oeis4.)