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

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

Views

Author

Michael Somos, Jul 05 2015

Keywords

Comments

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).

Examples

			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 + ...

Links

Crossrefs

Cf. A259655, A259825.

Programs

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

Formula

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).

A259657 Expansion of phi(-x^3) * f(-x^4)^3 / f(-x^12) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 0, -2, -3, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, -12, -9, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, -2, -12, 0, 0, 18, 0, 0, 0, 0, 6, 0, 0, -24, -12, 0, 0, 6, 0, 0, 0, 0, 20, 0, 0, -12, -12, 0, 0, 24, 0, 0, 0, 0, 24, 0, 0, -12, -21, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, -14, -24
Offset: 0

Views

Author

Michael Somos, Jul 02 2015

Keywords

Comments

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).

Examples

			G.f. = 1 - 2*x^3 - 3*x^4 + 6*x^7 + 8*x^12 - 12*x^15 - 9*x^16 + 6*x^19 + ...

Links

Crossrefs

Cf. A014453, A143161, A259655.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^4]^3 / QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[x^4]^3 / (QPochhammer[ -x^3, x^3]^2 QPochhammer[ -x^6, x^6]), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^3 / (eta(x^6 + A) * eta(x^12 + A)), n))};

Formula

Expansion of phi(-x^3) * b(x^4) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
Expansion of eta(q^3)^2 * eta(q^4)^3 / (eta(q^6) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, 0, -2, -3, 0, -1, 0, -3, -2, 0, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 72^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259655.
G.f.: Product_{k>0} (1 - x^(4*k))^3 / ((1 + x^(3*k))^2 * (1 + x^(6*k))).
a(3*n + 1) = -3 * A143161(n-1). a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(12*n) = A014453(n).

A259659 Expansion of phi(x^6) * f(-x)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 0, 6, -3, 0, 1, -9, 0, 12, -3, 0, 6, -12, 0, 6, -3, 0, 7, -15, 0, 18, -6, 0, 0, -15, 0, 24, -6, 0, 6, -15, 0, 6, -9, 0, 7, -21, 0, 30, -3, 0, 6, -21, 0, 24, -6, 0, 12, -27, 0, 0, -9, 0, 12, -21, 0, 36, -6, 0, 1, -18, 0, 36, -12, 0, 6, -33, 0, 18, -9, 0
Offset: 0

Views

Author

Michael Somos, Jul 02 2015

Keywords

Comments

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).

Examples

			G.f. = 1 - 3*x + 6*x^3 - 3*x^4 + x^6 - 9*x^7 + 12*x^9 - 3*x^10 + 6*x^12 + ...
G.f. = q^3 - 3*q^7 + 6*q^15 - 3*q^19 + q^27 - 9*q^31 + 12*q^39 - 3*q^43 + ...

Links

Crossrefs

Cf. A227595, A259655.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] QPochhammer[ x]^3 / QPochhammer[ x^3], {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-3/4)* eta[q]^3*eta[q^12]^2/(eta[q^3]*eta[q^6]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 17 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^6 + A)), n))};

Formula

Expansion of phi(x^6) * b(x) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
Expansion of q^(-3/4) * eta(q)^3 * eta(q^12)^2 / (eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -3, -3, -2, -3, -3, -1, -3, -3, -2, -3, -3, -3, ...].
a(2*n + 1) = -3 * A227595(n). a(3*n + 1) = -3 * A259655(n). a(3*n + 2) = 0.
Showing 1-4 of 4 results.

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

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