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A137566 Expansion of f(-x, -x^5) / f(-x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%I A137566 #7 Feb 16 2025 08:33:07
%S A137566 1,-1,0,0,0,-1,2,-2,1,0,0,-2,5,-5,2,0,1,-5,10,-10,5,-1,2,-10,20,-20,
%T A137566 10,-2,5,-20,36,-36,20,-6,10,-36,65,-65,36,-12,21,-65,110,-110,65,-25,
%U A137566 38,-110,185,-185,110,-46,70,-185,300,-300,186,-85,120,-300,481
%N A137566 Expansion of f(-x, -x^5) / f(-x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.
%H A137566 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A137566 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A137566 Expansion of q^(1/6) * eta(q) / (eta(q^2) * eta(q^3)) in powers of q.
%F A137566 Euler transform of period 6 sequence [ -1, 0, 0, 0, -1, 1, ...].
%F A137566 G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007690.
%F A137566 G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)))^(-1).
%e A137566 G.f. = 1 - x - x^5 + 2*x^6 - 2*x^7 + x^8 - 2*x^11 + 5*x^12 - 5*x^13 + 2*x^14 + ...
%e A137566 G.f. = 1/q - q^5 - q^29 + 2*q^35 - 2*q^41 + q^47 - 2*q^65 + 5*q^71 - 5*q^77 + ...
%t A137566 a[ n_] := SeriesCoefficient[ QPochhammer[ x] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* _Michael Somos_, Oct 12 2015 *)
%o A137566 (PARI) {a(n) = my(A); if( n<0, 0, A =x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A) / eta(x^3 + A), n))};
%Y A137566 Cf. A007690.
%K A137566 sign
%O A137566 0,7
%A A137566 _Michael Somos_, Jan 26 2008