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Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.
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%I #12 Mar 12 2021 22:24:47

%S 1,-2,0,0,2,0,0,0,2,-6,0,0,4,0,0,0,2,-4,0,0,0,0,0,0,4,-2,0,0,0,0,0,0,

%T 2,-8,0,0,6,0,0,0,0,-4,0,0,4,0,0,0,4,-2,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,

%U 2,0,0,0,4,0,0,0,6,-4,0,0,4,0,0,0,0,-10,0

%N Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A242609/b242609.txt">Table of n, a(n) for n = 0..2500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">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 eta(q)^2 * eta(q^16)^5 / (eta(q^2) * eta(q^8)^2 * eta(q^32)^2) in powers of q.

%F G.f.: (Sum_{k in Z} (-x)^k^2) * (Sum_{k in Z} (x^8)^k^2).

%F a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0. a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = -2 * A112603(n). a(8*n + 4) = 2 * A113411(n).

%F a(n) = (-1)^n * A226225(n).

%e G.f. = 1 - 2*q + 2*q^4 + 2*q^8 - 6*q^9 + 4*q^12 + 2*q^16 - 4*q^17 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];

%o (PARI) {a(n) = if( n<1, n==0, 2 * (-1)^n * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^16 + A)^5 / (eta(x^2 + A) * eta(x^8 + A)^2 * eta(x^32 + A)^2), n))};

%Y Cf. A033715, A112603, A113411, A226225.

%K sign

%O 0,2

%A _Michael Somos_, May 19 2014