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A280339 Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%I A280339 #11 Feb 16 2025 08:33:39
%S A280339 1,2,-1,-2,-5,-14,4,12,5,40,0,-26,11,-68,-15,30,-18,106,3,-50,-10,
%T A280339 -182,29,104,10,270,11,-130,37,-360,-51,164,-16,506,-30,-266,-65,-686,
%U A280339 62,320,53,898,22,-414,50,-1206,-61,612,-52,1560,-4,-696,-81,-1958,120
%N A280339 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 A280339 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A280339 G. C. Greubel, <a href="/A280339/b280339.txt">Table of n, a(n) for n = 0..5000</a>
%H A280339 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 A280339 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A280339 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A280339 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 A280339 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 A280339 Euler transform of period 8 sequence [2, -4, 2, -8, 2, -4, 2, -4, ...].
%F A280339 a(n) = (-1)^n * A279955(n).
%F A280339 a(3*n + 1) / a(1) == A138515(n) (mod 3). a(3^3*n + 7) / a(7) == A138515(n) (mod 3^2).
%e A280339 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 A280339 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 A280339 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 QPochhammer[ x]^2 QPochhammer[ -x^2, x^4]^4, {x, 0, n}];
%o A280339 (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 A280339 Cf. A138515, A279955.
%K A280339 sign
%O A280339 0,2
%A A280339 _Michael Somos_, Dec 31 2016