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A216711
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Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions.
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2
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1, 8, 12, -64, -210, 96, 1016, 512, -2043, -1680, 1092, -768, 1382, 8128, -2520, -4096, 14706, -16344, -39940, 13440, 12192, 8736, 68712, 6144, -34025, 11056, -50760, -65024, -102570, -20160, 227552, 32768, 13104, 117648, -213360, 130752, 160526, -319520
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q.
a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t).
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EXAMPLE
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G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}];
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))};
(Magma) A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* Michael Somos, Jun 10 2015 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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