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A137830 Expansion of phi(-x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A137830 #13 Feb 16 2025 08:33:07
%S A137830 1,-2,0,0,4,-4,0,0,9,-12,0,0,20,-24,0,0,42,-50,0,0,80,-92,0,0,147,
%T A137830 -172,0,0,260,-296,0,0,445,-510,0,0,744,-840,0,0,1215,-1372,0,0,1944,
%U A137830 -2176,0,0,3059,-3424,0,0,4740,-5268,0,0,7239,-8040,0,0,10920
%N A137830 Expansion of phi(-x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A137830 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A137830 G. C. Greubel, <a href="/A137830/b137830.txt">Table of n, a(n) for n = 0..1000</a>
%H A137830 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A137830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A137830 Expansion of q^(1/3) * eta(q)^2 / (eta(q^2) * eta(q^4)^2) in powers of q.
%F A137830 Euler transform of period 4 sequence [ -2, -1, -2, 1, ...].
%F A137830 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (4/3)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137829.
%F A137830 G.f.: ( Product_{k>0} (1 - x^(2*k)) * (1 + x^k)^2 * (1 + x^(2*k))^2 )^(-1).
%F A137830 a(4*n + 2) = a(4*n + 3) = 0.
%F A137830 a(n) = (-1)^n * A137828(n). a(4*n) = A051136(n). a(4*n + 1) = -2 * A137829(n).
%e A137830 G.f. = 1 - 2*x + 4*x^4 - 4*x^5 + 9*x^8 - 12*x^9 + 20*x^12 - 24*x^13 + 42*x^16 + ...
%e A137830 G.f. = 1/q - 2*q^2 + 4*q^11 - 4*q^14 + 9*q^23 - 12*q^26 + 20*q^35 - 24*q^38 + ...
%t A137830 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* _Michael Somos_, Oct 04 2015 *)
%o A137830 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A) / eta(x^4 + A)^2, n))};
%Y A137830 Cf. A051136, A137828, A137829.
%K A137830 sign
%O A137830 0,2
%A A137830 _Michael Somos_, Feb 12 2008