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A145706 Expansion of chi(-x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

%I A145706 #14 Feb 16 2025 08:33:09
%S A145706 1,0,1,0,1,-1,2,-1,2,-1,3,-2,4,-2,5,-4,6,-5,8,-6,11,-8,13,-10,16,-14,
%T A145706 20,-17,24,-21,31,-26,37,-32,44,-41,54,-49,64,-59,79,-72,94,-86,111,
%U A145706 -106,132,-126,156,-149,187,-178,219,-210,257,-251,302,-295,352
%N A145706 Expansion of chi(-x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.
%C A145706 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A145706 G. C. Greubel, <a href="/A145706/b145706.txt">Table of n, a(n) for n = 0..1000</a>
%H A145706 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A145706 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A145706 Expansion of q^(1/8) * eta(q^4) * eta(q^5) / (eta(q^2) * eta(q^10)) in powers of q.
%F A145706 Euler transform of period 20 sequence [ 0, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, ...].
%F A145706 G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).
%F A145706 G.f.: Product_{k>0} (1 - x^(10*k - 5)) / (1 - x^(4*k - 2)).
%F A145706 a(n) = (-1)^n * A139631(n) = A145704(2*n) = A145705(2*n).
%e A145706 G.f. = 1 + x^2 + x^4 - x^5 + 2*x^6 - x^7 + 2*x^8 - x^9 + 3*x^10 - 2*x^11 + ...
%e A145706 G.f. = 1/q + q^15 + q^31 - q^39 + 2*q^47 - q^55 + 2*q^63 - q^71 + 3*q^79 + ...
%t A145706 a[ n_] := SeriesCoefficient[ QPochhammer[ x^5, x^10] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)
%o A145706 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))};
%Y A145706 Cf. A139631, A145704, A145705.
%K A145706 sign
%O A145706 0,7
%A A145706 _Michael Somos_, Oct 17 2008, Oct 20 2008