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1, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,1
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COMMENTS
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This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^4, b = -x^3.
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LINKS
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FORMULA
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Expansion of f(-x^5, -x^16) - x * f(-x^2, -x^19) = f(-x^7, -x^14) * f(-x, -x^6) / f(-x^3, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 7 sequence [-1, 0, 1, 1, 0, -1, -1, ...].
G.f.: Sum_{k in Z} (-1)^k * x^(7*k * (3*k + 1) / 2) * (x^(9*k + 3) + x^(-9*k)).
G.f.: Product_{k>0} (1 - x^(7*k)) * (1 - x^(7*k - 1)) * (1 - x^(7*k - 6)) / ((1 - x^(7*k - 3)) * (1 - x^(7*k - 4))).
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EXAMPLE
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G.f. = 1 - x + x^3 - x^5 - x^16 + x^20 - x^26 + x^31 + x^53 - x^60 + x^70 + ...
G.f. = q^121 - q^289 + q^625 - q^961 - q^2809 + q^3481 - q^4489 + q^5329 + q^9025 + ...
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MATHEMATICA
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f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185175[n_] := SeriesCoefficient[f[-x^7, -x^14]*f[-x, -x^6]/ f[-x^3, -x^4], {x, 0, n}]; Table[A185175[n], {n, 0, 50}] (* G. C. Greubel, Jun 19 2017 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(7*k)) * (1 - x^(7*k-1)) * (1 - x^(7*k-6)) / ((1 - x^(7*k-3)) * (1 - x^(7*k-4))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 20 2017 *)
a[ n_] := With[{m = Sqrt[168 n + 121]}, If[ IntegerQ@m, KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Jun 27 2017 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, 0, -1, -1, 0, 1, 1}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jun 27 2017 *)
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PROG
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(PARI) {a(n) = my(m); if( issquare( 168*n + 121, &m), kronecker( 12, m))};
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[1, 1, 0, -1, -1, 0, 1][k%7+1]), n))}; /* Michael Somos, Jun 27 2017 */
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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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