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A144558 Expansion of Product_{n >= 1} (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))). 3

%I

%S 1,1,-1,0,3,2,-3,-1,8,5,-8,-3,18,11,-19,-7,38,22,-41,-16,75,42,-82,

%T -33,142,78,-157,-64,258,138,-288,-120,455,239,-511,-215,781,404,-882,

%U -374,1310,668,-1486,-635,2153,1084,-2450,-1053,3477,1733,-3967,-1712,5524,2726,-6316,-2737,8652,4233,-9907

%N Expansion of Product_{n >= 1} (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))).

%C The authors of the article have informed me that there is a typo in the published g.f. - the factor (1+q^(4*n-2)) should be squared. When this is done, we get the sequence A085261. In short, this is an erroneous version of A085261.

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

%H G. C. Greubel, <a href="/A144558/b144558.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Ishikawa and J. Zeng, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.064">The Andrews-Stanley partition function and Al-Salam-Chihara polynomials</a>, Disc. Math., 309 (2009), 151-175.

%H M. Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F From _Michael Somos_, Jun 04 2012: (Start)

%F Expansion of chi(x) / f(x^2) in powers of x where chi(), f() are Ramanujan theta functions.

%F Expansion of q^(1/8) * eta(q^2)^3 * eta(q^8) / (eta(q) * eta(q^4)^4) in powers of q.

%F Euler transform of period 8 sequence [ 1, -2, 1, 2, 1, -2, 1, 1, ...]. (End)

%e G.f. = 1 + x - x^2 + 3*x^4 + 2*x^5 - 3*x^6 - x^7 + 8*x^8 + 5*x^9 - 8*x^10 + ...

%e G.f. = 1/q + q^7 - q^15 + 3*q^31 + 2*q^39 - 3*q^47 - q^55 + 8*q^63 + 5*q^71 + ...

%t QP = QPochhammer; s = QP[q^2]^3*(QP[q^8]/(QP[q]*QP[q^4]^4)) + O[q]^60; CoefficientList[s, q] (* _Jean-Fran├žois Alcover_, Nov 14 2015, adapted from PARI *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^8 + A) / (eta(x + A) * eta(x^4 + A)^4), n))}; /* _Michael Somos_, Jun 04 2012 */

%K sign

%O 0,5

%A _N. J. A. Sloane_, Jan 02 2009

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Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)