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Expansion of f(-x^3)^6 / (phi(-x) * phi(-x^3)) in powers of x where phi(), f() are Ramanujan theta functions.
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%I #9 Mar 12 2021 22:24:48

%S 1,2,4,4,6,8,9,10,8,14,14,16,16,16,20,18,22,24,21,26,28,28,28,24,36,

%T 34,36,38,32,32,40,42,44,36,46,56,43,50,40,52,54,56,54,42,60,62,64,64,

%U 56,66,56,72,70,56,74,74,76,72,64,80,81,84,84,64,76,88,88

%N Expansion of f(-x^3)^6 / (phi(-x) * phi(-x^3)) 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="/A263021/b263021.txt">Table of n, a(n) for n = 0..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 q^(-3/4) * eta(q^2) * eta(q^3)^4 * eta(q^6) / eta(q)^2 in powers of q.

%F Euler transform of period 6 sequence [ 2, 1, -2, 1, 2, -4, ...].

%F a(3*n) = A261445(n). a(3*n + 1) = 2 * A260518(n). a(3*n + 2) = 4 * A260295(n).

%e G.f. = 1 + 2*x + 4*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 8*x^8 + ...

%e G.f. = q^3 + 2*q^7 + 4*q^11 + 4*q^15 + 6*q^19 + 8*q^23 + 9*q^27 + 10*q^31 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^6 / (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^2, n))};

%Y Cf. A260295, A260518, A261445.

%K nonn

%O 0,2

%A _Michael Somos_, Oct 07 2015