A143066 Expansion of phi(x^3) / psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
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
Keywords
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- S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 10th equation.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
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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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
Comments