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A133563
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Expansion of chi(-q) / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
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11
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1, -1, 0, -1, 1, 0, 0, -1, 1, -1, 2, -2, 2, -2, 2, -1, 2, -3, 2, -3, 5, -5, 4, -5, 6, -4, 4, -7, 7, -7, 10, -11, 10, -12, 12, -10, 12, -15, 14, -16, 22, -22, 20, -24, 26, -22, 24, -30, 31, -33, 40, -43, 42, -46, 48, -45, 50, -58, 58, -63, 77, -79, 76, -86, 92, -86, 92, -107, 110, -116, 134, -141, 142, -154, 160, -157
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OFFSET
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0,11
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COMMENTS
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In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015
Denoted by t in Andrews and Berndt 2005. - Michael Somos, Apr 25 2016
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REFERENCES
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G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.
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LINKS
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FORMULA
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Expansion of q^(-1/6) * eta(q) * eta(q^10) / ( eta(q^2) * eta(q^5) ) in powers of q.
Euler transform of period 10 sequence [ -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (360 t)) = f(t) where q = exp(2 Pi i t).
Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) + w^2 * (u^2 + v).
Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(x^q), B(q^9)) where f(u, v, w) = (u^3 + w^3) * (v + v^3) + 2 * v^4 - v^2 + u^3 * w^3 * ( 2 - v^2 ).
Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^5), B(q^10)) where f(u1, u2, u5, u10) = u1^2 * u5^2 + u1^2 * u10^4 + u1 * u2^2 * u5 * u10^2 + u2 * u5^2 * u10^3 + u2^3 * u10^3 - u2^2 * u10^2 - u1^3 * u5^3 - u1^4 * u10^2 - u1^3 * u2^2 * u5 - u1^2 * u2 * u5^2 * u10.
G.f.: Product_{k>0} P10(x^k) where P10 is the 10th cyclotomic polynomial.
G.f.: Product_{k>0} (1 + x^(5*k)) / (1 + x^k).
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/15)) / (2^(5/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015
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EXAMPLE
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G.f. = 1 - x - x^3 + x^4 - x^7 + x^8 - x^9 + 2*x^10 - 2*x^11 - 2*x^13 + ...
G.f. = q - q^7 - q^19 + q^25 - q^43 + q^49 - q^55 + 2*q^61 - 2*q^67 + 2*q^73 - ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^5, x^10], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A)), n))};
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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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