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A243747
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Expansion of (phi(q) - phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.
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2
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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, 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, 0, 2, 0, 0, 1, 4, -4, -2, 2, 0, 0, 0, -1, 0, 4, -2, -2, 0, 0, 0
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OFFSET
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2,2
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COMMENTS
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Expansion of third basis element of modular forms space for Gamma_1(8) of weight 1 in powers of q.
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LINKS
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FORMULA
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Expansion of (q * f(-q, -q^7)^2 / psi(-q))^2 in powers of q where psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [-2, 0, 2, 2, 2, 0, -2, -2, ...].
G.f.: (theta_3(x) - theta_3(x^2))^2 / 4 = (Sum_{k>0} x^(k^2) - x^(2k^2))^2.
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EXAMPLE
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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 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];
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PROG
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(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))))};
(Sage) ModularForms( Gamma1(8), 1, prec=70).2
(Magma) Basis( ModularForms( Gamma1(8), 1), 70) [3];
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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