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A134746 Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.

%I A134746 #28 Feb 16 2025 08:33:07
%S A134746 1,4,0,-16,0,56,0,-160,0,404,0,-944,0,2072,0,-4320,0,8648,0,-16720,0,
%T A134746 31360,0,-57312,0,102364,0,-179104,0,307672,0,-519808,0,864960,0,
%U A134746 -1419456,0,2299832,0,-3682400,0,5831784,0,-9141808,0,14194200,0,-21842368,0
%N A134746 Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
%C A134746 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A134746 G. C. Greubel, <a href="/A134746/b134746.txt">Table of n, a(n) for n = 0..1000</a>
%H A134746 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A134746 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A134746 Expansion of (phi(q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
%F A134746 Expansion of (eta(q^8) / eta(q))^4 * (eta(q^2) / eta(q^4))^14 in powers of q.
%F A134746 Euler transform of period 8 sequence [ 4, -10, 4, 4, 4, -10, 4, 0, ...].
%F A134746 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A210066.
%F A134746 G.f.: ( (Sum_{k in Z} x^(k^2)) / (Sum_{k in Z} x^(2*k^2)) )^2 = ( Product_{k>0} (1 + x^k)^2 * (1 + x^(4*k))^2 / (1 + x^(2*k))^5 )^2.
%F A134746 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u * (2 - u) * v^2.
%F A134746 a(2*n) = 0 unless n=0. a(2*n + 1) = 4 * A001938(n) = A127393(n).
%F A134746 a(n) = (-1)^n * A210067(n). Convolution inverse of A210066. - _Michael Somos_, Oct 16 2015
%F A134746 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 34 + 24*sqrt(2) - 4*sqrt(140 + 99*sqrt(2)). - _Simon Plouffe_, Mar 04 2021
%e A134746 G.f. = 1 + 4*q - 16*q^3 + 56*q^5 - 160*q^7 + 404*q^9 - 944*q^11 + 2072*q^13 + ...
%t A134746 CoefficientList[Series[(QPochhammer[x^8]/QPochhammer[x])^4 (QPochhammer[x^2]/QPochhammer[x^4])^14, {x, 0, 50}], x] (* _Jan Mangaldan_, Mar 21 2013 *)
%t A134746 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* _Michael Somos_, Oct 16 2015 *)
%o A134746 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7 )^2, n))};
%Y A134746 Cf. A001938, A127393, A210066, A210067.
%K A134746 sign
%O A134746 0,2
%A A134746 _Michael Somos_, Nov 07 2007