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.

A261426 Expansion of f(-x^3)^3 * phi(x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 4, 2, 5, 2, 6, 2, 6, 0, 4, 4, 7, 2, 4, 0, 6, 1, 8, 4, 4, 4, 10, 2, 8, 2, 12, 4, 8, 5, 6, 0, 14, 2, 8, 2, 11, 6, 6, 4, 8, 2, 8, 4, 8, 6, 14, 0, 6, 0, 12, 6, 15, 4, 14, 2, 14, 4, 8, 8, 12, 7, 14, 0, 12, 2, 16, 10, 8, 4, 10, 6, 14, 0, 16, 4, 16
Offset: 0

Views

Author

Michael Somos, Aug 18 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 + 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. A033687, A261394, A261426, A261444.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 3, 0, x^6] / QPochhammer[ x], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))};

Formula

Expansion of (1/3) * q^(-1/3) * c(q) * phi(q^6) in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM function. - Michael Somos, Sep 01 2015
Expansion of q^(-1/3) * eta(q^3)^3 * eta(q^12)^5 / (eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -5, 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (128/3)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261426.
24 * a(n) = A261394(6*n + 2).
a(n) = A261444(2*n). Michael Somos, Sep 01 2015

A263527 Expansion of phi(-x^3) * f(-x^6)^3 / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, -2, 2, -2, 0, -4, 2, 0, 1, -4, 4, -2, 2, -4, 5, 0, 2, -2, 6, -4, 2, -4, 6, 0, 0, -6, 4, -2, 4, -8, 7, 0, 2, -10, 4, -6, 0, -4, 6, 0, 1, -6, 8, -6, 4, -8, 4, 0, 4, -8, 10, -4, 2, -8, 8, 0, 2, -6, 12, -4, 4, -8, 8, 0, 5, -8, 6, -4, 0, -8, 14, 0, 2, -10
Offset: 0

Views

Author

Michael Somos, Oct 19 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 - 2*x^3 + 2*x^4 - 2*x^5 - 4*x^7 + 2*x^8 + x^10 - 4*x^11 + ...
G.f. = q^2 + q^8 - 2*q^11 + 2*q^14 - 2*q^17 - 4*q^23 + 2*q^26 + q^32 + ...

Links

Crossrefs

Cf. A261426, A261444, A263433, A263501.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^6]^3 / QPochhammer[ x^2], {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^6 + A)^2 / eta(x^2 + A), n))};

Formula

Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^6)^2 / eta(q^2) in powers of q.
Euler transform of period 6 sequence [ 0, 1, -2, 1, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (2048/3)^(1/2) (t/I)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263501.
a(n) = (-1)^n * A261444(n). a(8*n + 1) = 0.
a(2*n) = A261426(n). a(4*n) = A263433(n). a(4*n + 2) = A261444(n).

A263452 Expansion of f(-q^3)^3 * psi(q^12) / f(-q) in powers of q where ps(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 3, 1, 4, 0, 5, 3, 2, 0, 3, 3, 4, 0, 4, 2, 4, 0, 3, 2, 4, 0, 4, 2, 4, 0, 5, 5, 4, 0, 3, 3, 8, 0, 7, 3, 6, 0, 4, 4, 4, 0, 6, 4, 4, 0, 9, 3, 6, 0, 4, 4, 4, 0, 4, 3, 8, 0, 5, 5, 6, 0, 9, 3, 4, 0, 7, 6, 6, 0, 7, 6, 10, 0, 6, 3
Offset: 0

Views

Author

Michael Somos, Oct 18 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 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q^11 + q^17 + 2*q^23 + 2*q^35 + q^41 + 2*q^47 + q^59 + 2*q^65 + ...

Links

Crossrefs

Cf. A212907, A261444, A263527.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 EllipticTheta[ 2, 0, q^6] / ( 2 q^(3/2) QPochhammer[ q]), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};

Formula

Expansion of q^(-11/6) * eta(q^3)^3 * eta(q^24)^2 / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -1, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -3, ...].
2 * a(n) = A261444(2*n + 1). a(4*n + 1) = A212907(n). a(4*n + 3) = 0.
-2 * a(n) = A263527(2*n + 3). - Michael Somos, Nov 05 2015

A263502 Expansion of phi(q) * f(-q^2)^3 / f(-q^6) in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -3, -6, 2, 0, 0, 12, -3, -4, 12, -6, -6, 0, -6, 0, 2, -6, -12, 12, 0, 0, 24, -12, 0, 14, -6, -6, 12, 0, -24, 12, -3, 0, 12, -12, -4, 0, -12, -24, 12, -6, 0, 36, -6, 0, 24, -12, -6, 14, -15, 0, 0, 0, 0, 24, -6, -24, 36, -6, 0, 0, -18, -24, 2, -12, -24, 36
Offset: 0

Views

Author

Michael Somos, Oct 19 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36* t)) = 17496^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261444.

Examples

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

Links

Crossrefs

Cf. A261444, A263456.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] QPochhammer[ q^2]^3 / QPochhammer[ q^6], {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)), n))};

Formula

Expansion of eta(q^2)^8 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [2, -6, 2, -4, 2, -5, 2, -4, 2, -6, 2, -3, ...].
a(n) = A263456(4*n). a(8*n + 5) = a(9*n + 6) = 0.
a(3*n + 2) = -3 * A261444(n).
Showing 1-4 of 4 results.

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