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A135211 Expansion of psi(-x) / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

%I A135211 #19 Feb 16 2025 08:33:07
%S A135211 1,-1,0,0,-1,0,1,-1,0,2,-1,0,2,-2,0,2,-3,0,3,-3,0,4,-4,0,5,-6,0,6,-7,
%T A135211 0,7,-8,0,10,-10,0,13,-13,0,14,-16,0,17,-18,0,22,-22,0,26,-28,0,30,
%U A135211 -33,0,36,-38,0,44,-45,0,52,-55,0,60,-65,0,70,-74,0,84,-87,0,99,-104,0,112,-121,0,131,-138,0,156,-160
%N A135211 Expansion of psi(-x) / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.
%C A135211 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A135211 G. C. Greubel, <a href="/A135211/b135211.txt">Table of n, a(n) for n = 0..1000</a>
%H A135211 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A135211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A135211 Expansion of f(-x^2, -x^4) / f(x, x^5) in powers of x where f() is Ramanujan's two-variable theta function. - _Michael Somos_, Apr 05 2015
%F A135211 Expansion of q^(1/4) * eta(q) * eta(q^4) * eta(q^6) / ( eta(q^2) * eta(q^3) * eta(q^12) ) in powers of q.
%F A135211 Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, ...].
%F A135211 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 + v^4) - (1 + u*v)^3.
%F A135211 G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036018.
%F A135211 a(n) = (-1)^n * A256626(n). a(3*n + 1) = - A036018(n). a(3*n + 2) = 0. - _Michael Somos_, Apr 05 2015
%F A135211 Convolution inverse is A036018. Convolution square is A062243. Convolution 4th power is A187147. - _Michael Somos_, Apr 05 2015
%e A135211 G.f. = 1 - x -x^4 + x^6 - x^7 + 2*x69 - x^10 + 2*x^12 - 2*x^13 + 2*x^15 + ...
%e A135211 G.f. = 1/q - q^3 - q^15 + q^23 - q^27 + 2*q^35 - q^39 + 2*q^47 - 2*q^51 + ...
%t A135211 a[ n_] := SeriesCoefficient[ q^(1/4) EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)], {q, 0, n}]; (* _Michael Somos_, Apr 05 2015 *)
%o A135211 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
%Y A135211 Cf. A036018, A062243, A187147, A256626.
%K A135211 sign
%O A135211 0,10
%A A135211 _Michael Somos_, Nov 22 2007, Nov 23 2007