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A245572
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Expansion of phi(q) * phi(q^2) + 2 * phi(-q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.
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2
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3, 2, -2, 4, 6, 0, -4, 0, 6, 6, 0, 4, 12, 0, 0, 0, 6, 4, -6, 4, 0, 0, -4, 0, 12, 2, 0, 8, 0, 0, 0, 0, 6, 8, -4, 0, 18, 0, -4, 0, 0, 4, 0, 4, 12, 0, 0, 0, 12, 2, -2, 8, 0, 0, -8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 6, 0, -8, 4, 12, 0, 0, 0, 18, 4, 0, 4, 12, 0, 0, 0, 0
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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a(n) = 2 * b(n) where b(n) is multiplicative with b(2) = -1, b(2^e) = 3 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226240.
a(2*n + 1) = 2 * A113411(n). a(4*n) = 3 * A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n = 3) = 4 * A033761(n). a(8*n + 5) = a(8*n = 7) = 0.
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EXAMPLE
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G.f. = 3 + 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 4*q^6 + 6*q^8 + 6*q^9 + 4*q^11 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + 2 EllipticTheta[ 3, 0, -q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}];
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PROG
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(PARI) {a(n) = my(A, p, e); if( n<1, 3*(n==0), A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, if( e>1, 3, -1), p%8>3, (1 + (-1)^e) / 2, e+1)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^3 / (eta(x + A)^2 * eta(x^8 + A)^2) + 2 * eta(x^2 + A)^2 * eta(x^8 + A)^5 / (eta(x^4 + A)^3 * eta(x^16 + A)^2), n))};
(Magma) A := Basis( ModularForms( Gamma1(32), 1), 81); 3*A[1] + 2*A[2] - 2*A[3] + 4*A[4] + 6*A[5] - 4*A[7] + 6*A[9] + 6*A[10] + 4*A[12] + 12*A[13] + 4*A[16];
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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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