cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

Original entry on oeis.org

-1, 0, 0, 4, 6, 0, 0, 12, 12, 0, 0, 12, 16, 0, 0, 24, 18, 0, 0, 12, 24, 0, 0, 36, 24, 0, 0, 16, 24, 0, 0, 36, 36, 0, 0, 24, 30, 0, 0, 48, 24, 0, 0, 12, 48, 0, 0, 60, 40, 0, 0, 24, 24, 0, 0, 48, 48, 0, 0, 36, 48, 0, 0, 60, 42, 0, 0, 12, 48, 0, 0, 84, 36, 0, 0
Offset: 0

Views

Author

Michael Somos, Jul 05 2015

Keywords

Comments

Coefficients of q-expansion of Eisenstein series G_{3/2}(tau) multiplied by 12. - N. J. A. Sloane, Mar 16 2019

Examples

			G.f. = -1 + 4*x^3 + 6*x^4 + 12*x^7 + 12*x^8 + 12*x^11 + 16*x^12 + 24*x^15 + ...

Links

Crossrefs

Cf. A004024, A005869, A005876, A005877, A005878, A005884, A005886, A008443.
Cf. A045828, A045831, A045834, A058305, A058306, A130695, A181648, A185220.
Cf. A188569, A213022, A213617, A213624, A213625, A213627, A259827.
See also A306934-A306937.

Programs

  • 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^(2 k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; gf[terms // Sqrt // Ceiling] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2017 *)
a[ n_] := If[ n<1, -Boole[n==0], With[{m = Floor[(-1 + Sqrt[1 + 4*n])/2]}, -2*SeriesCoefficient[ Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, -m-1,m}] / EllipticTheta[3, 0, x] + Sum[(-1)^k*x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, -m-2,m}]/ EllipticTheta[3, 0, -x], {x, 0, n}]]]; (* Michael Somos, Feb 04 2022 *)
  • PARI
    {a(n) = 12 * qfbhclassno(n)};
  • PARI
    {a(n) = my(D, f); 12 * if( n<1, (n==0)/-12, [D, f] = core(-n, 1); if( D%4>1 && !(f%2), D*=4; f/=2); if( D%4<2, qfbclassno(D) / max(1, D+6), 0) * sumdiv(f, d, moebius(d) * kronecker(D, d) * sigma(f/d)))};

Formula

a(n) = 12 * A058305(n) / A058306(n). a(4*n + 1) = a(4*n + 2) = 0. a(3*n + 4) = 6 * A259827(n).
a(4*n + 3) = 4 * A130695(n). a(8*n + 3) = A005886(n) = 2 * A005869(n) = 4 * A008443(n). a(12*n + 7) = 12 * A259655(n).
a(16*n + 4) = 6 * A045834(n) = 3 * A005876(n). a(16*n + 8) = 12 * A045828(n) = 6 * A005884(n) = 3 * A005877(n).
a(24*n + 3) = 4 * A213627(n). a(24*n + 7) = 12 * A185220(n). a(24*n + 11) = 12 * A213617(n). a(24*n + 19) = 12 * A181648(n). a(24*n + 23) = 12 * A188569(n+1).
a(32*n + 4) = 6 * A213022(n). a(32*n + 8) = 12 * A213625(n). a(32*n + 12) = 16 * A008443(n) = 8 * A005869(n) = 4 * A005886(n) = 2 * A005878(n). a(32*n + 20) = 24 * A045831(n) = 6 * A004024(n). a(32*n + 24) = 24 * A213624(n).
G.f.: -2 * (Sum_{k in Z} (-1)^k * x^(k*k + k) / (1 + (-x)^k)^2) / (Sum_{k in Z} x^k^2) - 2 * (Sum_{k in Z} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (Sum_{k in Z} (-x)^k^2).
a(n) >= 0 if n > 0. - Michael Somos, Feb 04 2022

A306934 Coefficients of q-expansion of Eisenstein series G_{5/2}(tau) multiplied by 120.

Original entry on oeis.org

1, -10, 0, 0, -70, -48, 0, 0, -120, -250, 0, 0, -240, -240, 0, 0, -550, -480, 0, 0, -528, -480, 0, 0, -720, -1210, 0, 0, -960, -720, 0, 0, -1080, -1440, 0, 0, -1750, -1200, 0, 0, -1680, -1920, 0, 0, -1680, -1488, 0, 0, -2160, -3370, 0, 0, -2640, -1680, 0, 0, -2400, -3360, 0, 0, -2880, -2640
Offset: 0

Views

Author

N. J. A. Sloane, Mar 16 2019

Keywords

Links

Crossrefs

Cf. A259825, A306935, A306936, A306937.

Programs

  • Sage
    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*sigma(f/d, 3) for d in f.divisors()])
    return round((120*X.lfunction(100)(-1)*s).real()) # Robin Visser, Feb 24 2024

Extensions

Corrected and more terms from Robin Visser, Feb 24 2024

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

Views

Author

N. J. A. Sloane, Mar 16 2019

Keywords

Links

Crossrefs

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

Views

Author

N. J. A. Sloane, Mar 16 2019

Keywords

Links

Crossrefs

Cf. A259825, A306934, A306935, A306937.

Programs

  • Sage
    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

Extensions

More terms from Robin Visser, Feb 24 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.