A058305 Numerator of H(n), where H(0)=-1/12, H(n) = number of equivalence classes of positive definite quadratic forms a*x^2+b*x*y+c*y^2 with discriminant b^2-4ac = -n, counting forms equivalent to x^2+y^2 (resp. x^2+x*y+y^2) with multiplicity 1/2 (resp. 1/3).
-1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 4, 0, 0, 2, 3, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 4, 2, 0, 0, 3, 3, 0, 0, 2, 5, 0, 0, 4, 2, 0, 0, 1, 4, 0, 0, 5, 10, 0, 0, 2, 2, 0, 0, 4, 4, 0, 0, 3, 4, 0, 0, 5, 7, 0, 0, 1, 4, 0, 0, 7, 3, 0, 0, 7, 4, 0, 0, 5, 6, 0, 0, 3, 4, 0, 0, 6, 2, 0, 0, 2, 6, 0, 0, 8, 6, 0, 0, 3
Offset: 0
Examples
-1/12, 0, 0, 1/3, 1/2, 0, 0, 1, 1, ...
References
- D. Zagier, The Eichler-Selberg Trace Formula on SL_2(Z), Appendix to S. Lang, Introduction to Modular Forms, Springer, 1976.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
- N. Lygeros, O. Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
- Wikipedia, Hurwitz class number.
Programs
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Mathematica
terms = 100; gf[m_] := With[{r = Range[-m, m]}, -2 Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, r}]/EllipticTheta[3, 0, x] - 2 Sum[(-1)^k*x^(k^2 + 2 k)/(1 + x^(2k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; CoefficientList[ gf[terms // Sqrt // Ceiling] + O[x]^terms, x]/12 // Numerator (* Jean-François Alcover, Apr 02 2017, after Michael Somos *) -
PARI
H(n)=sumdiv(core(n,1)[2],d,my(D=-n/d^2);if(D%4<2,qfbclassno(D)/max(1,D+6))) a(n)=if(n,numerator(H(n)),-1) \\ Charles R Greathouse IV, Apr 25 2013
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PARI
{a(n) = numerator( qfbhclassno( n))}; /* Michael Somos, Jul 06 2015 */
Formula
H(n) = A259825(n) / 12. - Michael Somos, Jul 05 2015
Comments