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.

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A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 15, 24, 33, 36, 33, 48, 69, 78, 90, 72, 51, 84, 120, 144, 141, 108, 87, 120, 198, 192, 180, 144, 87, 186, 210, 240, 264, 180, 198, 192, 285, 288, 270, 288, 105, 228, 300, 336, 414, 252, 264, 264, 396, 468, 360, 288, 159, 342, 465, 432, 462, 324, 249
Offset: 0

Views

Author

Michael Somos, Sep 19 2013

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).
Number 8 and 32 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

Examples

			G.f. = 1 + 6*q + 15*q^2 + 24*q^3 + 33*q^4 + 36*q^5 + 33*q^6 + 48*q^7 + 69*q^8 + ...

Links

Crossrefs

Cf. A107760, A227226.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 6*A[2] + 15*A[3];
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^2]^6 / (QPochhammer[q]^3 QPochhammer[q^6]^2))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^6 / EllipticTheta[ 2, 0, q^3]^2/16, {q, 0, 2 n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 3/2, 2, 3/2, 2, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 1, 2, 1, 2, -8}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2))^2, n))};
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 6*A[1] + 15*A[2];

Formula

Expansion of (a(q) + a(q^2))^2 / 4 in powers of q where a() is a cubic AGM theta function.
Expansion of (b(q^2)^2 / b(q))^2 in powers of q where b() is a cubic AGM theta function.
Expansion of (eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2))^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227226.
Convolution square of A107760.
Euler transform of period 6 sequence [6, -6, 4, -6, 6, -4, ...].

A321466 Expansion of (phi(x^3)^3 / phi(x))^2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 12, -20, 28, -24, 28, -32, 60, -68, 72, -48, 44, -56, 96, -120, 124, -72, 76, -80, 168, -160, 144, -96, 76, -124, 168, -212, 224, -120, 168, -128, 252, -240, 216, -192, 92, -152, 240, -280, 360, -168, 224, -176, 336, -408, 288, -192, 140, -228, 372
Offset: 0

Views

Author

Michael Somos, Nov 11 2018

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).
The g.f. of A113973 is k = k(q) := phi(x^3)^3 / phi(x) given in equation (2.2) page 996 of Williams 2012, and the g.f. of k^2 which is given in equation (2.3) page 997 is this sequence.
Number 54 of the 126 eta-quotients listed in Table 1 of Williams 2012.

Examples

			G.f. = 1 - 4*x + 12*x^2 - 20*x^3 + 28*x^4 - 24*x^5 + 28*x^6 + ...

Links

Crossrefs

Cf. A113973,A227226,A321465.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 2), 51); A[1] - 4*A[2] + 12*A[3] - 20*A[4] + 28*A[5];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3]^6 / EllipticTheta[ 3, 0, x]^2, {x, 0, n}];
a[ n_] := With[{s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n == 0], -4 (s[n] - 6 s[n/2] + s[n/3] + 4 s[n/4] + 2 s[n/6] + 4 s[n/12])]];
  • PARI
    {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -4*(s(n) - 6*s(n/2) + s(n/3) + 4*s(n/4) + 2*s(n/6) + 4*s(n/12)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^15 / (eta(x^2 + A)^5 * eta(x^3 + A)^6 * eta(x^12 + A)^6))^2, n))};

Formula

Expansion of (eta(q)^2 * eta(q^4)^2 * eta(q^6)^15 / (eta(q^2)^5 * eta(q^3)^6 * eta(q^12)^6))^2 in powers of q.
Expansion of ((a(x) - 2*a(x^2) - 2*a(x^4))/3)^2 = ((b(x) + 2*b(x^4))^2 / (9*b(x^2)))^2 in powers of x where a(), b() are cubic AGM theta functions.
Euler transform of period 12 sequence [-4, 6, 8, 2, -4, -12, -4, 2, 8, 6, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321465.
G.f.: (theta_3(0, x^3)^3 / theta_3(0, x))^2 where theta_3(0, x) is a Jacobi theta function.
G.f.: (Product_{k>0} f(x^k))^2 where f(x) := ((1 - x) * (1 + x^2))^2 * ((1 - x^3) * (1 + x^3)^3)^3 / ((1 - x^2) * (1 + x^6)^2)^3.
a(n) = -4*(s(n) - 6*s(n/2) + s(n/3) + 4*s(n/4) + 2*s(n/6) + 4*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(n) = (-1)^n * A227226(n). Convolution square of A113973.
Showing 1-2 of 2 results.

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