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 1-2 of 2 results.

A260574 Expansion of phi(x^3) * psi(x^3) / f(x) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 2, 0, 2, -1, 4, -2, 5, -2, 6, -2, 10, -4, 12, -4, 15, -6, 20, -8, 26, -9, 32, -12, 40, -16, 50, -18, 60, -22, 76, -28, 92, -33, 110, -40, 134, -50, 160, -58, 191, -70, 230, -84, 272, -98, 320, -116, 380, -138, 446, -160, 522, -188, 612, -222, 715, -256
Offset: 0

Views

Author

Michael Somos, Jul 29 2015

Keywords

Comments

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

Examples

			G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 4*x^6 - 2*x^7 + 5*x^8 - 2*x^9 + ...
G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 4*q^19 - 2*q^22 + 5*q^25 + ...

Links

Crossrefs

Cf. A085140, A097196, A257655.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3]^3 / (QPochhammer[ -x] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^3 * eta(x^12 + A)^2), n))};
  • PARI
    q='q+O('q^99); Vec(eta(q)*eta(q^4)*eta(q^6)^7/(eta(q^2)^3*eta(q^3)^3*eta(q^12)^2)) \\ Altug Alkan, Aug 01 2018

Formula

Expansion of f(x^3)^3 / (f(x) * phi(-x^6)) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 1, -1, -2, -1, 1, 2, 2, -1, -1, ...].
a(2*n) = A085140(n). a(2*n + 1) = - A097196(n). a(4*n + 1) = - A257655(n).

A257655 Expansion of f(x^3, x^9) * f(x^6, x^6) / f(-x, -x^2) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 138, 188, 256, 348, 463, 614, 812, 1060, 1378, 1785, 2292, 2932, 3740, 4736, 5978, 7522, 9416, 11750, 14620, 18116, 22384, 27585, 33878, 41500, 50714, 61794, 75120, 91118, 110247, 133110, 160390, 192836, 231400, 277162
Offset: 0

Views

Author

Michael Somos, Jul 25 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 22*x^7 + 33*x^8 + ...
G.f. = q + q^4 + 2*q^7 + 4*q^10 + 6*q^13 + 9*q^16 + 16*q^19 + 22*q^22 + ...

Links

Crossrefs

Cf. A097196.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8) QPochhammer[ x]), {x, 0, n}];
eta[q_] := q^(1/24)*QPochhammer[q]; With[{nmax = 50}, CoefficientList[ Series[q^(-1/3)*eta[q^12]^5/(eta[q]*eta[q^3]*eta[q^24]^2), {x, 0, nmax}], x]] (* G. C. Greubel, Aug 02 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^12 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^24 + A)^2), n))};

Formula

Expansion of f(-x, -x^5) * f(x^6, x^6) / f(-x, -x) in powers of x where f(,) is Ramanujan's general theta function.
Expansion of q^(-1/3) * eta(q^12)^5 / (eta(q) * eta(q^3) * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, -3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, -1, ...].
a(n) = A097196(2*n).
Showing 1-2 of 2 results.

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