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%I #14 Mar 12 2021 22:24:44
%S 1,1,0,2,2,0,0,0,2,3,0,2,4,0,0,0,2,2,0,2,0,0,0,0,4,1,0,4,0,0,0,0,2,4,
%T 0,0,6,0,0,0,0,2,0,2,4,0,0,0,4,1,0,4,0,0,0,0,0,4,0,2,0,0,0,0,2,0,0,2,
%U 4,0,0,0,6,2,0,2,4,0,0,0,0,5,0,2,0,0,0
%N Expansion of (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 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="/A129438/b129438.txt">Table of n, a(n) for n = 0..1000</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 Moebius transform is period 32 sequence [1, -1, 1, 2, -1, -1, -1, 0, 1, 1, 1, 2, -1, 1, -1, 0, 1, -1, 1, -2, -1, -1, -1, 0, 1, 1, 1, -2, -1, 1, -1, 0, ...].
%F a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0.
%F a(n) = A125096(n) unless n=0. a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n).
%F a(2*n + 1) = A113411(n). a(4*n) = A033715(n). - _Michael Somos_, Nov 11 2015
%e G.f. = 1 + q + 2*q^3 + 2*q^4 + 2*q^8 + 3*q^9 + 2*q^11 + 4*q^12 + 2*q^16 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + EllipticTheta[ 4, 0, q^2] EllipticTheta[ 3, 0, q^4]) / 2, {q, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])};
%Y Cf. A033715, A033761, A112603, A113411, A125096.
%K nonn
%O 0,4
%A _Michael Somos_, Apr 14 2007
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