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Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
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%I #10 Mar 12 2021 22:24:48

%S 1,2,-1,-2,-5,-14,4,12,5,40,0,-26,11,-68,-15,30,-18,106,3,-50,-10,

%T -182,29,104,10,270,11,-130,37,-360,-51,164,-16,506,-30,-266,-65,-686,

%U 62,320,53,898,22,-414,50,-1206,-61,612,-52,1560,-4,-696,-81,-1958,120

%N Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), 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="/A280339/b280339.txt">Table of n, a(n) for n = 0..5000</a>

%H Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016)

%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 phi(-x^4)^2 * chi(-x^4)^2 * f(x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%F Expansion of q^(1/4) * eta(q^2)^6 * eta(q^4)^4 / (eta(q)^2 * eta(q^8)^4) in powers of q.

%F Euler transform of period 8 sequence [2, -4, 2, -8, 2, -4, 2, -4, ...].

%F a(n) = (-1)^n * A279955(n).

%F a(3*n + 1) / a(1) == A138515(n) (mod 3). a(3^3*n + 7) / a(7) == A138515(n) (mod 3^2).

%e G.f. = 1 + 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 + 4*x^6 + 12*x^7 + 5*x^8 + ...

%e G.f. = q^-1 + 2*q^3 - q^7 - 2*q^11 - 5*q^15 - 14*q^19 + 4*q^23 + 12*q^27 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 QPochhammer[ x]^2 QPochhammer[ -x^2, x^4]^4, {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^4 + A)^4 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

%Y Cf. A138515, A279955.

%K sign

%O 0,2

%A _Michael Somos_, Dec 31 2016