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Expansion of q^(1/4) * (eta(q^8) * eta(q^10) + eta(q^2) * eta(q^40)) / (eta(q^4) * eta(q^20)) in powers of q.
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%I #12 Mar 05 2018 04:10:20

%S 1,1,0,-1,1,0,0,-1,1,1,-1,-1,2,1,-1,-1,2,2,-1,-2,3,3,-2,-3,4,3,-2,-4,

%T 5,4,-4,-5,6,6,-5,-6,8,7,-6,-8,11,10,-8,-11,13,11,-10,-13,16,15,-14,

%U -17,20,18,-17,-20,24,23,-21,-25,31,29,-26,-32,37,34,-32

%N Expansion of q^(1/4) * (eta(q^8) * eta(q^10) + eta(q^2) * eta(q^40)) / (eta(q^4) * eta(q^20)) in powers of q.

%C Denoted by "(160~b)" in Simon Norton's replicable function list.

%H G. C. Greubel, <a href="/A145704/b145704.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = (-1)^n * A145705(n). a(2*n) = A145706(n). a(2*n + 1) = A145707(n).

%e G.f. = 1 + x - x^3 + x^4 - x^7 + x^8 + x^9 - x^10 - x^11 + 2*x^12 + x^13 + ...

%e G.f. = 1/q + q^3 - q^11 + q^15 - q^27 + q^31 + q^35 - q^39 - q^43 + 2*q^47 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^8] QPochhammer[ x^10] + x QPochhammer[ x^2] QPochhammer[ x^40]) / (QPochhammer[ x^4] QPochhammer[ x^20]), {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^10 + A) + x * eta(x^2 + A) * eta(x^40 + A)) / (eta(x^4 + A) * eta(x^20 + A)), n))};

%Y Cf. A145705, A145706, A145707.

%K sign

%O 0,13

%A _Michael Somos_, Oct 17 2008, Nov 11 2008, Jan 21 2009