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%I #19 Mar 12 2021 22:24:46
%S 1,-1,0,-1,1,-1,1,-1,2,-2,2,-2,3,-3,3,-4,5,-5,5,-6,7,-8,8,-9,11,-11,
%T 11,-14,15,-16,17,-19,22,-23,24,-27,31,-32,34,-38,42,-44,47,-52,57,
%U -61,64,-70,78,-82,87,-96,103,-110,117,-127,138,-146,155,-168,182
%N Expansion of q * chi(-q) / chi(-q^25) 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="/A213419/b213419.txt">Table of n, a(n) for n = 1..1000</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 Expansion of (eta(q) * eta(q^50)) / (eta(q^2) * eta(q^25)) in powers of q.
%F Euler transform of period 50 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (50 t)) = f(t) where q = exp(2 Pi i t).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v - u^2) * (v - w^2) - 2*u*w * (1 + w^2).
%F G.f.: x * (Product_{k>0} (1 + x^(25*k)) / (1 + x^k)).
%F Convolution inverse of A034320.
%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)/5) / (2*sqrt(5)*n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e G.f. = q - q^2 - q^4 + q^5 - q^6 + q^7 - q^8 + 2*q^9 - 2*q^10 + 2*q^11 - 2*q^12 + ...
%t QP = QPochhammer; s = (QP[q]*QP[q^50])/(QP[q^2]*QP[q^25]) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^25, q^25] / QPochhammer[ -q, q], {q, 0, n}]; (* _Michael Somos_, May 05 2016 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^50 + A) / (eta(x^2 + A) * eta(x^25 + A)), n))};
%Y Cf. A034320.
%K sign
%O 1,9
%A _Michael Somos_, Jun 11 2012
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