A259825 -id:A259825 - cp's OEIS frontend

A058305 Numerator of H(n), where H(0)=-1/12, H(n) = number of equivalence classes of positive definite quadratic forms ax^2+bx*y+cy^2 with discriminant b^2-4ac = -n, counting forms equivalent to x^2+y^2 (resp. x^2+xy+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

N. J. A. Sloane, Dec 09 2000

H(n) is usually called the Hurwitz class number.

a(n) = 0 if n = 1 or 2 (mod 4).

			-1/12, 0, 0, 1/3, 1/2, 0, 0, 1, 1, ...

D. Zagier, The Eichler-Selberg Trace Formula on SL_2(Z), Appendix to S. Lang, Introduction to Modular Forms, Springer, 1976.

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.

Cf. A058306, A259825.

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 *)

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

{a(n) = numerator( qfbhclassno( n))}; /* Michael Somos, Jul 06 2015 */

H(n) = A259825(n) / 12. - Michael Somos, Jul 05 2015

A058306 Denominator of H(n), where H(0)=-1/12, H(n) = number of equivalence classes of positive definite quadratic forms ax^2+bx*y+cy^2 with discriminant b^2-4ac = -n, counting forms equivalent to x^2+y^2 (resp. x^2+xy+y^2) with multiplicity 1/2 (resp. 1/3).

Original entry on oeis.org

12, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Dec 09 2000

H(n) is usually called the Hurwitz class number.

a(n) = 1 unless n is of the form 3k^2 or 4k^2. - Charles R Greathouse IV, Apr 25 2013

			-1/12, 0, 0, 1/3, 1/2, 0, 0, 1, 1, ...

D. Zagier, The Eichler-Selberg Trace Formula on SL_2(Z), Appendix to S. Lang, Introduction to Modular Forms, Springer, 1976.

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.

Cf. A058305, A259825.

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 // Denominator (* Jean-François Alcover, Apr 02 2017, after Michael Somos *)

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,denominator(H(n)),12) \\ Charles R Greathouse IV, Apr 25 2013

a(n)=if(n,my(D=4-valuation(n,3)%2);denominator(if(issquare(n/D) && n%D==0, qfbclassno(-D)/max(1,6-D))),12) \\ Charles R Greathouse IV, Apr 25 2013

{a(n) = denominator( qfbhclassno( n))}; /* Michael Somos, Jul 06 2015 */

H(n) = A259825(n) / 12. - Michael Somos, Jul 05 2015

A259827 Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 4, 2, 0, 0, 4, 8, 0, 0, 7, 2, 0, 0, 8, 10, 0, 0, 4, 4, 0, 0, 5, 10, 0, 0, 8, 4, 0, 0, 12, 10, 0, 0, 8, 6, 0, 0, 4, 14, 0, 0, 12, 2, 0, 0, 8, 14, 0, 0, 8, 4, 0, 0, 9, 18, 0, 0, 12, 6, 0, 0, 16, 14, 0, 0, 4, 4, 0, 0, 12, 12

Offset: 0

Michael Somos, Jul 05 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 4*x^12 + 2*x^13 + ...
G.f. = q^4 + 2*q^7 + 3*q^16 + 2*q^19 + 4*q^28 + 6*q^31 + 4*q^40 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A259655, A259825.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x^12]^3 / QPochhammer[ x^4], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};

Expansion of phi(x) * c(x^4) / (3 * x^(4/3)) in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
Expansion of q^(-4/3) * eta(q^2)^5 * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 2, 0, 2, -3, 2, 0, 2, -3, 2, -3, ...].
a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A259655(n). 6 * a(n) = A259825(3*n + 4).

A297122 a(n) = n^5*H(n) where H() is the Hurwitz class number.

Original entry on oeis.org

0, 0, 0, 81, 512, 0, 0, 16807, 32768, 0, 0, 161051, 331776, 0, 0, 1518750, 1572864, 0, 0, 2476099, 6400000, 0, 0, 19309029, 15925248, 0, 0, 19131876, 34420736, 0, 0, 85887453, 100663296, 0, 0, 105043750, 151165440, 0, 0, 360896796, 204800000, 0, 0, 147008443

Offset: 0

Seiichi Manyama, Dec 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000594, A259825, A297127.

PARI
```
{a(n) = n^5*qfbhclassno(n)}
```

a(n) = n^5*A259825(n)/12.

a(4*n+1) = a(4*n+2) = 0.

A306935 Coefficients of q-expansion of Eisenstein series G_{7/2}(tau) multiplied by -252.

Original entry on oeis.org

1, 0, 0, 56, 128, 0, 0, 576, 756, 0, 0, 1512, 2072, 0, 0, 4032, 4158

Offset: 0

N. J. A. Sloane, Mar 16 2019

D. Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Cf. A259825, A306934-A306937.

A306936 Coefficients of q-expansion of Eisenstein series G_{9/2}(tau) multiplied by 240.

Original entry on oeis.org

1, 2, 0, 0, 242, 480, 0, 0, 2640, 4322, 0, 0, 11040, 13920, 0, 0, 30962, 39360, 0, 0, 65760, 73920, 0, 0, 125280, 156002, 0, 0, 216960, 226080, 0, 0, 340560, 406080, 0, 0, 522962, 541920, 0, 0, 756960, 860160, 0, 0, 1033440, 1063200, 0, 0, 1424160, 1646402, 0, 0, 1907040, 1860000, 0, 0

Offset: 0

N. J. A. Sloane, Mar 16 2019

nonn

H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285.
X. Wang and D. Pei, Modular Forms with Integral and Half-Integral Weights, Science Press Beijing, Springer Berlin Heidelberg, 2012. x+432 pp.
D. Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Cf. A259825, A306934, A306935, A306937.

def a(n):
    if n==0: return 1
    if (n%4) not in [0, 1]: return 0
    D = Integer(n).squarefree_part()
    f = Integer(sqrt(n/D))
    if (D%4) not in [0, 1]: D, f = 4*D, f//2
    X = kronecker_character(D)
    s = sum([moebius(d)*X(d)*d^3*sigma(f/d, 7) for d in f.divisors()])
    return round((240*X.lfunction(100)(-3)*s).real()) # Robin Visser, Feb 24 2024

More terms from Robin Visser, Feb 24 2024

A260195 Number of integer triples [x, y, z] such that 1 <= min(x,z), max(x,z) <= y, y^2 - (x^2 - x + z^2 - z) / 2 = n.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 4, 3, 6, 4, 3, 5, 6, 4, 9, 5, 3, 7, 7, 5, 9, 6, 6, 8, 9, 5, 9, 8, 6, 10, 6, 5, 15, 8, 9, 10, 7, 7, 12, 10, 3, 11, 15, 7, 15, 8, 6, 13, 12, 9, 12, 9, 9, 14, 12, 7, 15, 12, 6, 15, 13, 6, 21, 12, 12, 13, 6, 11, 15, 15, 9, 14, 12, 8, 24, 10, 9

Offset: 0

Michael Somos, Jul 18 2015

nonn

Same as A238872 except a(0) = 0.

			G.f. = x + x^2 + 3*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 3*x^8 + ...

Wikipedia, Hurwitz class number.

Cf. A130695, A238872, A259825.

a[ n_] := If[ OddQ[n], 1, 1/3] Length @ FindInstance[ {x >= 0, y >= 0, z >= 0, x y + y z + z x + x + y + z + 1 == n}, {x, y, z}, Integers, 10^9];
a[ n_] := Length @ FindInstance[ {1 <= y <= n, 1 <= x <= y, 1 <= z <= y, y^2 - (x^2 - x + z^2 - z) / 2 == n}, {x, y, z}, Integers, 10^9];

{a(n) = my(c, t, i); for(k=1 + sqrtint(max(0, n-1)), n, forstep(j=1, min(2*k, sqrtint(t = 8*k^2 - 8*n + 2)), 2, if( issquare( t - j^2, &i) && i<=2*k, c++))); c};

a(n) = A238872(n) unless n=0. a(2*n) = A130695(2*n) / 3. a(2*n + 1) = A130695(2*n + 1) = A259825(8*n + 3) / 4 = 3 * H(8*n + 3) where H() is the Hurwitz class number.

A369902 Number of isomorphism classes of elliptic curves over the finite field of order prime(n) whose trace of Frobenius is zero.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 4, 6, 6, 6, 2, 8, 4, 10, 6, 12, 6, 4, 14, 4, 10, 12, 12, 4, 14, 10, 12, 6, 8, 10, 20, 8, 12, 14, 14, 6, 4, 22, 14, 20, 10, 26, 4, 10, 18, 12, 14, 20, 10, 12, 30, 12, 28, 16, 26, 22, 22, 6, 20, 12, 18, 12, 38, 8, 10, 12, 8, 20, 14, 16, 38, 18, 10, 12, 34, 22, 6, 20, 16

Offset: 1

Robin Visser, Feb 05 2024

nonn

a(n) is the number of isomorphism classes of elliptic curves E over the finite field F_p such that E has exactly p+1 points over F_p.

			For n = 1, the unique a(1) = 1 elliptic curve over F_2 whose trace of Frobenius is zero is y^2 + y = x^3.
For n = 2, the a(2) = 2 elliptic curves over F_3 whose trace of Frobenius is zero are y^2 = x^3 + x and y^2 = x^3 + 2*x.
For n = 3, the a(3) = 2 elliptic curves over F_5 whose trace of Frobenius is zero are y^2 = x^3 + 1 and y^2 = x^3 + 2.

Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), no. 2, 183-211.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Cf. A259825, A362243.

a(n) = if (n<=2, n, qfbhclassno(4*prime(n)));

# A brute force computation of a(n)
def a(n):
    if n==1: return 1
    p, ECs = Primes()[n-1], []
    for A,B in ((x, y) for x in range(p) for y in range(p)):
        if ((4*A^3 + 27*B^2)%p != 0):
            E = EllipticCurve(GF(p), [A,B])
            if (E.trace_of_frobenius()==0):
                if not any([E.is_isomorphic(Ei) for Ei in ECs]): ECs.append(E)
    return len(ECs)

a(n) = A259825(4*prime(n))/12 if n > 2.

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A058306 Denominator 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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A259827 Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A297122 a(n) = n^5*H(n) where H() is the Hurwitz class number.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A306935 Coefficients of q-expansion of Eisenstein series G_{7/2}(tau) multiplied by -252.

Views

Author

Keywords

Links

Crossrefs

A306936 Coefficients of q-expansion of Eisenstein series G_{9/2}(tau) multiplied by 240.

Views

Author

Keywords

Links

Crossrefs

Programs

Sage

Extensions

A260195 Number of integer triples [x, y, z] such that 1 <= min(x,z), max(x,z) <= y, y^2 - (x^2 - x + z^2 - z) / 2 = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369902 Number of isomorphism classes of elliptic curves over the finite field of order prime(n) whose trace of Frobenius is zero.

Views

A058305 Numerator of H(n), where H(0)=-1/12, H(n) = number of equivalence classes of positive definite quadratic forms ax^2+bx*y+cy^2 with discriminant b^2-4ac = -n, counting forms equivalent to x^2+y^2 (resp. x^2+xy+y^2) with multiplicity 1/2 (resp. 1/3).

A058306 Denominator of H(n), where H(0)=-1/12, H(n) = number of equivalence classes of positive definite quadratic forms ax^2+bx*y+cy^2 with discriminant b^2-4ac = -n, counting forms equivalent to x^2+y^2 (resp. x^2+xy+y^2) with multiplicity 1/2 (resp. 1/3).