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%I #13 Mar 12 2021 22:24:44
%S 1,-1,0,-1,2,-1,0,-2,3,-2,2,-4,6,-5,4,-6,9,-8,6,-10,15,-14,12,-17,24,
%T -21,18,-26,35,-32,30,-42,52,-50,48,-60,75,-74,70,-88,111,-109,104,
%U -130,158,-154,150,-184,220,-218,218,-262,308,-308,308,-362,421,-426,428,-498,580,-589,592,-685,788,-796
%N Expansion of chi(q^5) * chi(q^10) / ( chi(q) * chi(q^2)) in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A128763/b128763.txt">Table of n, a(n) for n = 0..1500</a>
%H Michael 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 Euler transform of period 40 sequence [ -1, 0, -1, 1, 0, 0, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 0, -1, 0, ...].
%F Given g.f. A(x), then B(q) = 1/q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u^2 + v^2).
%F G.f.: Product_{k>0} (1 + x^(4*k)) * (1 + x^(5*k)) / ( (1 + x^k) * (1 + x^(20*k)) ).
%F Convolution inverse of A128762.
%F Expansion of q^(1/2)*eta(q)*eta(q^8)*eta(q^10)*eta(q^20)/(eta(q^2)* eta(q^4)*eta(q^5)*eta(q^40)) in powers of q. - _G. C. Greubel_, Jul 03 2018
%e G.f. = 1 - x - x^3 + 2*x^4 - x^5 - 2*x^7 + 3*x^8 - 2*x^9 + 2*x^10 - 4*x^11 + ...
%e G.f. = 1/q - q - q^5 + 2*q^7 - q^9 - 2*q^13 + 3*q^15 - 2*q^17 + 2*q^19 - ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x, -x] QPochhammer[ x^2, -x^2]) / (QPochhammer[ x^5, - x^5] QPochhammer[ x^10, -x^10]) , {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; A:= q^(1/2)*eta[q]*eta[q^8]* eta[q^10]*eta[q^20]/(eta[q^2]*eta[q^4]*eta[q^5] *eta[q^40]); a:= CoefficientList[Series[A, {q, 0, 80}], q]; Table[a[[n]], {n, 1, 80}] (* _G. C. Greubel_, Jul 03 2018 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8 + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^40 + A)), n))};
%Y Cf. A128762.
%K sign
%O 0,5
%A _Michael Somos_, Mar 25 2007
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