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A214303
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Expansion of f(-x^2, -x^4) * f(x^1, x^7) in powers of x where f(,) is Ramanujan's two-variable theta function.
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2
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1, 1, -1, -1, -1, -1, 0, 1, 0, -1, 2, 0, -1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, -1, -1, -1, 1, 0, -1, -1, -3, 1, 0, -1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 1, 2, -1, -1, 0, -1, 0, 0, 1, 1, 0, 1, -1, -1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, -1, -1, 0, 0, 0, 1, -1, 0, -1, -1
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OFFSET
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0,11
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COMMENTS
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a(n) = sum of (-1)^((u-1)/6) over all solutions of 48*n + 31 = 4*u^2 + 3*v^2 in integers where u == 1 (mod 6) and v == 3 (mod 8).
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LINKS
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FORMULA
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Euler transform of period 16 sequence [ 1, -2, 0, -1, 0, -1, 1, -2, 1, -1, 0, -1, 0, -2, 1, -2, ...].
G.f.: (Sum_{k} (-1)^k * x^(3*k^2 + k)) * (Sum_{k} x^(4*k^2 + 3*k)).
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EXAMPLE
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1 + x - x^2 - x^3 - x^4 - x^5 + x^7 - x^9 + 2*x^10 - x^12 + x^15 + x^17 + ...
q^31 + q^79 - q^127 - q^175 - q^223 - q^271 + q^367 - q^463 + 2*q^511 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^8] QPochhammer[ -q^1, q^8] QPochhammer[ -q^7, q^8], {q, 0, n}]
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PROG
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(PARI) {a(n) = local(s, v); if( n<0, 0, n = 48*n + 31; forstep( u=1, sqrtint( n\4), 2, if( u%3 && issquare( (n - 4*u^2)/3, &v), s += (-1)^((u+1)\6))); s)}
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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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