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Expansion of f(-x^2)^3 * f(-x^6)^3 / f(-x)^2 in powers of x where f() is a Ramanujan theta function.
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%I #30 Mar 12 2021 22:24:48

%S 1,2,2,4,5,6,7,6,9,8,11,14,10,14,15,16,14,14,19,20,21,22,21,20,28,26,

%T 24,22,29,30,26,32,26,38,35,36,37,28,39,40,41,42,34,40,43,42,47,42,49,

%U 50,56,44,42,54,55,62,57,46,50,60,56,62,50,70,60,56,74,54

%N Expansion of f(-x^2)^3 * f(-x^6)^3 / f(-x)^2 in powers of x where f() 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="/A260295/b260295.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^(-11/12) * eta(q^2)^3 * eta(q^6)^3 / eta(q)^2 in powers of q.

%F Euler transform of period 6 sequence [2, -1, 2, -1, 2, -4, ...]. - _Michael Somos_, Aug 01 2018

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

%e G.f. = q^11 + 2*q^23 + 2*q^35 + 4*q^47 + 5*q^59 + 6*q^71 + 7*q^83 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 QPochhammer[ x^6]^3 / QPochhammer[ 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)^3 * eta(x^6 + A)^3 / eta(x + A)^2, n))};

%o (PARI) q='q+O('q^99); Vec(eta(q^2)^3*eta(q^6)^3/eta(q)^2) \\ _Altug Alkan_, Jul 31 2018

%K nonn

%O 0,2

%A _Michael Somos_, Oct 07 2015