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A143066
Expansion of phi(x^3) / psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
4
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
0,6
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 10th equation.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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
EXAMPLE
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 + ...
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 *)
PROG
(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))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved