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 1-3 of 3 results.

A143068 Expansion of phi(q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 4, 8, 0, 4, 10, 0, 8, 16, 0, 8, 20, 0, 14, 30, 0, 16, 36, 0, 24, 52, 0, 28, 64, 0, 42, 88, 0, 48, 108, 0, 68, 144, 0, 80, 176, 0, 108, 230, 0, 128, 280, 0, 170, 360, 0, 200, 436, 0, 260, 552, 0, 308, 666, 0, 392, 832, 0
Offset: 0

Views

Author

Michael Somos, Jul 21 2008

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q + 2*q^4 + 2*q^6 + 4*q^7 + 2*q^9 + 4*q^10 + 4*q^12 + 8*q^13 + ...

Links

Crossrefs

Cf. A143066.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Formula

Expansion of eta(q^2)^5 * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^2 in powers of q.
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -1, 2, -1, 2, -3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143066.
a(3*n + 2) = 0.
G.f.: ( Sum_{k in Z} x^k^2 ) / ( Sum_{k in Z} (-x^6)^k^2 ).

A262150 Expansion of f(-x^3)^3 / (f(-x^2) * f(-x^4)^2) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 1, -3, 4, -3, 5, -12, 14, -10, 18, -37, 41, -34, 54, -98, 109, -92, 138, -237, 260, -230, 329, -531, 583, -526, 728, -1129, 1233, -1143, 1537, -2292, 2503, -2355, 3097, -4486, 4889, -4677, 6031, -8502, 9263, -8962, 11372, -15680, 17066, -16703, 20893
Offset: 0

Views

Author

Michael Somos, Sep 13 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 - 3*x^3 + 4*x^4 - 3*x^5 + 5*x^6 - 12*x^7 + 14*x^8 + ...
G.f. = q^-1 + q^47 - 3*q^71 + 4*q^95 - 3*q^119 + 5*q^143 - 12*q^167 + ...

Links

Crossrefs

Cf. A143066.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / (QPochhammer[ x^2] QPochhammer[ x^4]^2), {x, 0, n}];
nmax = 60; CoefficientList[Series[Product[(1-x^(3*k))^3 / ((1-x^(2*k))^3 * (1+x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / (eta(x^2 + A) * eta(x^4 + A)^2), n))};

Formula

Expansion of q^(1/24) * eta(q^3)^3 / (eta(q^2) * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [ 0, 1, -3, 3, 0, -2, 0, 3, -3, 1, 0, 0, ...].
a(n) = A143066(3*n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(3/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Sep 13 2015

A262929 Expansion of phi(-x^3) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 0, -2, -1, 0, 0, 2, 1, 0, 0, -2, 0, 0, 0, 4, 1, 0, 0, -6, -2, 0, 0, 8, 1, 0, 0, -12, -1, 0, 0, 16, 2, 0, 0, -22, -3, 0, 0, 30, 2, 0, 0, -38, -1, 0, 0, 50, 4, 0, 0, -66, -5, 0, 0, 84, 3, 0, 0, -106, -3, 0, 0, 136, 6, 0, 0, -172, -8, 0, 0, 214, 5, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 04 2015

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x^3 - x^4 + 2*x^7 + x^8 - 2*x^11 + 4*x^15 + x^16 + ...
G.f. = q^-1 - 2*q^5 - q^7 + 2*q^13 + q^15 - 2*q^21 + 4*q^29 + q^31 + ...

Links

Crossrefs

Cf. A143066, A261877, A262150, A262151, A262152, A262156, A262157, A262158, A262160.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2 x^(1/2) EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 2, 0, 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^4 + A) / (eta(x^6 + A) * eta(x^8 + A)^2), n))};
  • PARI
    q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)/(eta(q^6)*eta(q^8)^2)) \\ Altug Alkan, Jul 31 2018

Formula

Expansion of q^(1/2) * eta(q^3)^2 * eta(q^4) / (eta(q^6) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [0, 0, -2, -1, 0, -1, 0, 1, -2, 0, 0, -2, 0, 0, -2, 1, 0, -1, 0, -1, -2, 0, 0, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (32/3)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261877.
a(4*n) = A143066(n). a(4*n + 1) = a(4*n + 2) = 0. a(4*n + 3) = -2 * A262160(n).
a(12*n) = A262150(n). a(12*n + 3) = -2*A262151(n). a(12*n + 4) = -A262152(n). a(12*n + 7) = 2*A262156(n). a(12*n + 8) = A262157(n). a(12*n + 11) = -2*A262158(n). - Michael Somos, Apr 03 2016
Convolution inverse is A261877. - Michael Somos, Oct 22 2017
Showing 1-3 of 3 results.

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