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%I #14 Jul 27 2024 01:25:59
%S 1,4,15,44,121,300,707,1572,3366,6932,13865,26952,51187,95080,173280,
%T 310172,546438,948360,1623737,2744840,4585920,7577684,12393330,
%U 20073648,32219481,51270912,80927964,126758160,197096678,304339020,466829342,711555332,1078037580
%N Expansion of f(-x^6) / (f(-x^2)^3 * 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).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A259664/b259664.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 1 / (b(x^2) * phi(-x)^2) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
%F Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q)^4 * eta(q^2)) in powers of q.
%F Euler transform of period 6 sequence [ 4, 5, 4, 5, 4, 2, ...].
%F G.f.: Product_{k>0} (1 - x^(6*k))^3 / ((1 - x^k)^4 * (1 - x^(2*k))).
%F 3 * a(n) = A132974(2*n + 1). -3 * a(n) = A132979(2*n + 1).
%F a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (6^(9/4) * n^(5/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = 1 + 4*x + 15*x^2 + 44*x^3 + 121*x^4 + 300*x^5 + 707*x^6 + 1572*x^7 + ...
%e G.f. = q + 4*q^3 + 15*q^5 + 44*q^7 + 121*q^9 + 300*q^11 + 707*q^13 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^3 / (QPochhammer[ x]^4 QPochhammer[ x^2]), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^3 / (eta(x + A)^4 * eta(x^2 + A)), n))};
%Y Cf. A132974, A132979.
%K nonn
%O 0,2
%A _Michael Somos_, Jul 02 2015
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