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.

A143066 Expansion of phi(x^3) / psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 1, 0, 1, -2, 1, -1, 2, -3, 2, -1, 4, -5, 3, -3, 6, -8, 5, -4, 9, -12, 8, -7, 14, -18, 13, -10, 20, -26, 18, -16, 29, -37, 27, -23, 41, -52, 38, -34, 58, -72, 54, -47, 79, -98, 74, -67, 109, -133, 103, -92, 146, -178, 138, -127, 196, -237, 187, -170, 260
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 - x + x^2 + x^4 - 2*x^5 + x^6 - x^7 + 2*x^8 - 3*x^9 + 2*x^10 + ...
G.f. = 1/q - q^7 + q^15 + q^31 - 2*q^39 + q^47 - q^55 + 2*q^63 - 3*q^71 + ...

References

  • S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 10th equation.

Links

Crossrefs

Cf. A143068, A262150, A262152, A262157, A262929.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {-x^2}, x^2, x], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/8) EllipticTheta[ 3, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ x^(1/8)EllipticTheta[ 2, 0, x^(3/2)]^2 / (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3]), {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Formula

Expansion of q^(-1/8) * eta(q) * eta(q^6)^5 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [ -1, 1, 1, 1, -1, -2, -1, 1, 1, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (2/3)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143068.
G.f.: (1 + 2 * x^3 + 2 * x^12 + 2 * x^27 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan]
G.f.: 1 - x * (1 - x) / (1 - x^4) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - x^9 * (1 - x) * (1 - x^3) * (1 - x^5) / ((1 - x^4) * (1 - x^8) * (1 - x^12)) + ... [Ramanujan]
-psi6 +2*psi3 -psi1
Expansion of psi(x^3)^2 / (psi(x) * psi(x^6)) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 08 2015
a(n) = A262929(4*n). a(3*n) = A262150(n). a(3*n + 1) = - A262152(n). a(3*n + 2) = A262157(n). - Michael Somos, Nov 08 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

A262162 Expansion of f(-x^2)^5 * f(-x^12)^3 / (f(x)^2 * f(-x^8)^6) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 0, 4, 0, 0, 1, -12, 0, 0, -3, 30, 0, 0, 4, -66, 0, 0, -3, 136, 0, 0, 5, -268, 0, 0, -12, 506, 0, 0, 14, -920, 0, 0, -10, 1622, 0, 0, 18, -2788, 0, 0, -37, 4688, 0, 0, 41, -7726, 0, 0, -34, 12506, 0, 0, 54, -19928, 0, 0, -98, 31306, 0, 0, 109
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 - 2*x + 4*x^5 + x^8 - 12*x^9 - 3*x^12 + 30*x^13 + 4*x^16 + ...
G.f. = q^-1 - 2*q^5 + 4*q^29 + q^47 - 12*q^53 - 3*q^71 + 30*q^77 + ...

Links

Crossrefs

Cf. A262150, A262151.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^5 QPochhammer[ x^12]^3 / (QPochhammer[ -x]^2 QPochhammer[ x^8]^6), {x, 0, n}];
  • 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^12 + A)^3 / (eta(x^2 + A) * eta(x^8 + A)^6), n))};

Formula

Expansion of q^(1/6) * eta(q)^2 * eta(q^4)^2 * eta(q^12)^3 / (eta(q^2) * eta(q^8)^6) in powers of q.
Euler transform of period 24 sequence [ -2, -1, -2, -3, -2, -1, -2, 3, -2, -1, -2, -6, -2, -1, -2, 3, -2, -1, -2, -3, -2, -1, -2, 0, ...].
a(4*n) = A262150(n). a(4*n + 1) = -2 * A262151(n). a(4*n + 2) = a(4*n + 3) = 0.
Showing 1-3 of 3 results.

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