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A246838 Expansion of f(-x^2) * f(-x^12)^2 / f(x^1, x^5) in powers of x where f() is Ramanujan theta function.

%I A246838 #14 Feb 16 2025 08:33:23
%S A246838 1,-1,0,0,-1,0,1,-1,0,2,-1,0,0,0,0,2,-1,0,1,-1,0,0,-2,0,0,-1,0,2,0,0,
%T A246838 0,-1,0,0,-1,0,3,-1,0,0,-1,0,2,-1,0,2,0,0,0,-1,0,0,-1,0,2,-1,0,0,0,0,
%U A246838 1,-2,0,0,-2,0,0,-1,0,2,-1,0,2,0,0,0,-1,0,0,0
%N A246838 Expansion of f(-x^2) * f(-x^12)^2 / f(x^1, x^5) in powers of x where f() is Ramanujan theta function.
%C A246838 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246838 G. C. Greubel, <a href="/A246838/b246838.txt">Table of n, a(n) for n = 0..1000</a>
%H A246838 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246838 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246838 Expansion of q^(-3/4) * eta(q) * eta(q^4) * eta(q^6) * eta(q^12) / (eta(q^2) * eta(q^3)) in powers of q.
%F A246838 Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, -2, ...].
%F A246838 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 27^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246752.
%F A246838 a(3*n) = A112604(n). a(3*n + 1) = - A121361(n). a(3*n + 2) = 0.
%e A246838 G.f. = 1 - x - x^4 + x^6 - x^7 + 2*x^9 - x^10 + 2*x^15 - x^16 + x^18
%e A246838 + ...
%e A246838 G.f. = q^3 - q^7 - q^19 + q^27 - q^31 + 2*q^39 - q^43 + 2*q^63 - q^67 + ...
%t A246838 a[ n_] := SeriesCoefficient[ x^(1/4) EllipticTheta[ 2, Pi/4, x^(1/2)] QPochhammer[ x^12]^2 / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* _Michael Somos_, Aug 27 2015 *)
%o A246838 (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^12 + A) / (eta(x^2 + A) * eta(x^3 + A)), n))};
%Y A246838 Cf. A112604, A121361, A246752.
%K A246838 sign
%O A246838 0,10
%A A246838 _Michael Somos_, Sep 04 2014