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 A138559 Expansion of q^(1/24) * eta(q^2)^4 / (eta(q^4)^2 * eta(q)) in powers of q. 1

%I

%S 1,1,-2,-1,1,-1,-1,-1,2,2,-2,0,1,1,-1,0,3,1,-3,-2,3,0,-2,-1,3,2,-4,-2,

%T 2,1,-4,-2,5,3,-6,-1,5,1,-5,-3,6,3,-6,-3,7,2,-6,-2,9,5,-10,-5,9,3,-9,

%U -4,11,6,-12,-4,11,5,-12,-5,14,6,-16,-7,15,5,-16,-7,19,9,-20,-8,19,7,-20,-10,24,11,-25,-11,24,9,-26,-11,29,13,-31,-13

%N Expansion of q^(1/24) * eta(q^2)^4 / (eta(q^4)^2 * eta(q)) in powers of q.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of phi(q) * chi(-q) = phi(-q^2) * chi(q) = psi(q) * chi(-q^2)^2 = f(-q) * chi(q)^2 = f(q) * chi(-q^2) = f(q)^2 / psi(q) = phi(-q^2)^2 / f(-q) where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F Euler transform of period 4 sequence [ 1, -3, 1, -1, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A096920.

%F G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2.

%e 1/q + q^23 - 2*q^47 - q^71 + q^95 - q^119 - q^143 - q^167 + 2*q^191 + ...

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x^4 + A)^2 / eta(x + A), n))}

%K sign

%O 0,3

%A Michael Somos, Mar 24 2008

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