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%I #9 Mar 12 2021 22:24:48
%S 1,2,-5,-10,9,14,-10,0,14,2,-11,-32,0,14,-9,26,2,0,16,-22,14,0,0,26,
%T -17,-32,-22,-10,-34,14,45,38,0,-34,38,-22,2,0,-10,64,-20,0,0,0,-23,
%U -46,16,0,-46,-32,26,-10,25,18,0,38,50,0,0,-22,-80,50,0,26,2
%N Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A258779/b258779.txt">Table of n, a(n) for n = 0..2500</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 q^(-1/12) * (eta(q^2)^5 / (eta(q) * eta(q^4)^2))^2 in powers of q.
%F Euler transform of period 4 sequence [ 2, -8, 2, -4, ...].
%F a(n) = A000727(2*n) = A187076(2*n) = A106508(4*n) = A187149(4*n).
%F Convolution square of A143378.
%e G.f. = 1 + 2*x - 5*x^2 - 10*x^3 + 9*x^4 + 14*x^5 - 10*x^6 + 14*x^8 + ...
%e G.f. = q + 2*q^13 - 5*q^25 - 10*q^37 + 9*q^49 + 14*q^61 - 10*q^73 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] EllipticTheta[ 3, 0, x])^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};
%Y Cf. A000727, A106508, A143378, A187076, A187149.
%K sign
%O 0,2
%A _Michael Somos_, Jun 09 2015
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