A165684 -id:A165684 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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).

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

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

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 *)

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

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

A165682 Primes p such that 3p(p-1)+1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 43, 53, 59, 67, 89, 109, 137, 157, 173, 197, 239, 313, 347, 353, 389, 449, 521, 547, 557, 571, 577, 599, 613, 647, 677, 733, 743, 787, 857, 907, 941, 977, 1051, 1069, 1093, 1153, 1193, 1223, 1229, 1237, 1303, 1433, 1453, 1459, 1481, 1571

Offset: 1

Vincenzo Librandi, Sep 24 2009

			For p=5 = a(3), 3*p^2-3*p+1=61 = A165683(3). For a(4)=p=7: 3*p^2-3*p+1=127 = A165684(4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A165681, A165683.

[p: p in PrimesUpTo(1600) | IsPrime(3*p*(p-1)+1)]; // Vincenzo Librandi, Apr 15 2013

Select[Prime[Range[800]], PrimeQ[3 # (# - 1) + 1]&] (* Vincenzo Librandi, Apr 15 2013 *)

3*a(n)*(a(n)-1)+1 = A165683(n).

2 and 3 inserted by R. J. Mathar, Sep 26 2009

A165685 Dimension of the space of Siegel cusp forms of genus 2 and weight n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0, 7, 0, 8, 0, 9, 0, 11, 1, 13, 0, 13, 1, 17, 1, 18, 1, 20, 2, 23, 3, 26, 2, 27, 4, 32, 4, 34, 5, 37, 6, 41, 8, 46, 7, 47, 10, 54, 11, 57, 12, 61, 14, 67, 17, 73, 16, 75, 21, 84, 22, 88, 24, 94, 27, 101, 31, 109, 31, 112

Offset: 1

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

nonn

			a(35)=1 as the dimension of the space of Siegel cusp form of genus 2 and weight 35 is 1.

M. Eie, Dimensions of spaces of Siegel cusp forms of degree two and three, AMS, 1984, p. 44-45.

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Proof of generating function.

Cf. A008615, A029143. A165684 gives only the even weights.

N1[k_] := 2^(-7)*3^(-3)*5^(-1) (2 k^3 + 96 k^2 - 52 k - 3231) /; Mod[k, 2] == 0; N1[k_] := 2^(-7)*3^(-3)*5^(-1)*(2 k^3 - 114 k^2 + 2018 k - 9051) /; Mod[k, 2] == 1; N2[k_] := 2^(-5)*3^(-3)*(17 k - 294) /; Mod[k, 12] == 0; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 325) /; Mod[k, 12] == 1; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 254) /; Mod[k, 12] == 2; N2[k_] := 2^(-5)*3^(-3)*(17 k - 261) /; Mod[k, 12] == 3; N2[k_] := 2^(-5)*3^(-3)*(17 k - 86) /; Mod[k, 12] == 4; N2[k_] := 2^(-5)*3^(-3)*(-k + 53) /; Mod[k, 12] == 5; N2[k_] := 2^(-5)*3^(-3)*(-k - 42) /; Mod[k, 12] == 6; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 91) /; Mod[k, 12] == 7; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 2) /; Mod[k, 12] == 8; N2[k_] := 2^(-5)*3^(-3)*(-k - 27) /; Mod[k, 12] == 9;
N2[k_] := 2^(-5)*3^(-3)*(-k + 166) /; Mod[k, 12] == 10; N2[k_] := 2^(-5)*3^(-3)*(17 k - 181) /; Mod[k, 12] == 11; N3[k_] := 2^(-7)*3^(-3)*1131 /; Mod[k, 12] == 0; N3[k_] := 2^(-7)*3^(-3)*229 /; Mod[k, 12] == 1; N3[k_] := 2^(-7)*3^(-3)*(-229) /; Mod[k, 12] == 2; N3[k_] := 2^(-7)*3^(-3)*(-1131) /; Mod[k, 12] == 3; N3[k_] := 2^(-7)*3^(-3)*427 /; Mod[k, 12] == 4; N3[k_] := 2^(-7)*3^(-3)*(-571) /; Mod[k, 12] == 5;
N3[k_] := 2^(-7)*3^(-3)*123 /; Mod[k, 12] == 6; N3[k_] := 2^(-7)*3^(-3)*(-203) /; Mod[k, 12] == 7; N3[k_] := 2^(-7)*3^(-3)*203 /; Mod[k, 12] == 8; N3[k_] := 2^(-7)*3^(-3)*(-123) /; Mod[k, 12] == 9; N3[k_] := 2^(-7)*3^(-3)*571 /; Mod[k, 12] == 10; N3[k_] := 2^(-7)*3^(-3)*(-427) /; Mod[k, 12] == 11; N4[k_] := 5^(-1) /; Mod[k, 5] == 0; N4[k_] := -5^(-1) /; Mod[k, 5] == 3; N4[k_] := 0 /; Mod[k, 5] == 1 || Mod[k, 5] == 2 || Mod[k, 5] == 4;
DimSk[k_] := If[k >= 7, N1[k] + N2[k] + N3[k] + N4[k], 0];
Table[ DimSk[k], {k, 1, 100}]
(* second program: *)
init = {0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0};
ker = {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};
ans = LinearRecurrence[ker, init, 100];
ans[[3]] = 0 ; ans (* Andy Huchala, Mar 03 2022 *)

R. = PowerSeriesRing(ZZ, 100);
p = x^26 + x^24 - x^21 - x^19 + x^18 - x^17 - x^14 - x^13 + x^10 + x^9 + x^8 + x^7 - x^3;
q = x^27 - x^23 - x^22 - x^21 + x^18 + x^17 + x^16 - x^15 - x^12 + x^11 + x^10 + x^9 - x^6 - x^5 - x^4 + 1;
(x^3 + p/q).list()[1:] # Andy Huchala, Mar 03 2022

G.f.: x^10 (1+x^2-x^5-x^7+x^10-x^15+x^20) / ((-1+x)^4 (1+x)^3 (1+2x^2+2x^4+x^6)^2 (1+x+x^4+x^7+x^8)). - Andy Huchala, Mar 03 2022

a(2n) = A165684(n) and a(2n+35) = A029143(n). - Andy Huchala, Mar 04 2022

a(73) corrected by Andy Huchala, Mar 02 2022

A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 8, 11, 12, 14, 16, 19, 20, 24, 26, 29, 32, 37, 38, 44, 47, 51, 56, 62, 64, 72, 76, 82, 88, 96, 99, 109, 115, 122, 130, 140, 144, 157, 164, 173, 183, 195, 201, 216, 225, 236, 248, 263, 270, 288, 299, 312, 327, 344, 353, 374

Offset: 1

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

Also the dimension of the largest Hecke-closed subspace of forms in S_k(Gamma_2) which satisfy the Ramanujan-Petersson conjecture. These forms are also characterized by the property that their (Andrianov) spinor zeta function does not have any pole.

			a(20)=1 because there is exactly one Siegel modular form of genus 2 and weight 20 which is not a lift of some form of genus 1.

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhaeusser, 1985.
T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256(3), p. 323-340, 1981.
R. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)

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).

Cf. A165684 for the full space of Siegel cusp forms. See also A029143, A027640, A165685.

For k > 1 we have a(k) = A165684(k) - A008615(2k-5).

Conjectured G.f.: -x^10*(x^7+x^6-x^2-x-1) / ((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). - Colin Barker, Mar 30 2013

cp's OEIS Frontend

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.

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A165682 Primes p such that 3*p*(p-1)+1 is also prime.

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Magma

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A165685 Dimension of the space of Siegel cusp forms of genus 2 and weight n.

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Programs

Mathematica

Sage

Formula

Extensions

A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

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.

A165682 Primes p such that 3p(p-1)+1 is also prime.