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%I #11 Sep 08 2022 08:46:08
%S 1,-2,1,2,-2,0,1,-2,4,-2,-2,2,0,0,1,0,-1,-2,4,0,-2,0,-2,2,4,-4,0,2,0,
%T 0,1,-4,2,0,-1,2,-2,0,4,0,0,-2,-2,2,0,0,-2,0,5,-4,4,2,-4,0,0,-4,4,-2,
%U 0,2,0,0,1,4,-4,-2,2,0,0,0,-1,0,4,-2,-2,0,0,0
%N Expansion of (phi(q) - phi(q^2))^2 / 4 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).
%C Expansion of third basis element of modular forms space for Gamma_1(8) of weight 1 in powers of q.
%H G. C. Greubel, <a href="/A243747/b243747.txt">Table of n, a(n) for n = 2..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 (q * f(-q, -q^7)^2 / psi(-q))^2 in powers of q where psi(), f() are Ramanujan theta functions.
%F Euler transform of period 8 sequence [-2, 0, 2, 2, 2, 0, -2, -2, ...].
%F G.f.: (theta_3(x) - theta_3(x^2))^2 / 4 = (Sum_{k>0} x^(k^2) - x^(2k^2))^2.
%F Convolution square of A143259.
%e G.f. = q^2 - 2*q^3 + q^4 + 2*q^5 - 2*q^6 + q^8 - 2*q^9 + 4*q^10 - 2*q^11 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];
%o (PARI) {a(n) = if( n<2, 0, sum(k=1, n-1, (issquare(k) - issquare(2*k)) * (issquare(n - k) - issquare(2*n - 2*k))))};
%o (Sage) ModularForms( Gamma1(8), 1, prec=70).2
%o (Magma) Basis( ModularForms( Gamma1(8), 1), 70) [3];
%Y CF. A143259.
%K sign
%O 2,2
%A _Michael Somos_, Jun 09 2014
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