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A134131
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Expansion of chi(-x) * chi(-x^9) / chi(-x^3)^2 in power of x where chi() is a Ramanujan theta function.
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2
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1, -1, 0, 1, -1, -1, 2, -2, 0, 2, -2, -1, 4, -4, 0, 5, -4, -2, 8, -7, -1, 9, -8, -3, 14, -13, -2, 16, -14, -5, 24, -21, -3, 27, -24, -8, 39, -35, -6, 45, -39, -13, 62, -55, -10, 71, -62, -19, 96, -85, -16, 111, -96, -29, 146, -128, -25, 168, -146, -42, 218
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OFFSET
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0,7
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(1/6) * eta(q) * eta(q^6)^2 * eta(q^9) / (eta(q^2) * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, -1, 0, ...].
Given g.f. A(x) then B(q) = A(q^6) / q satisfies 0 = f(B(q), B(q^2), B(q^4) ) where f(u, v, w) = (u^2 + v) * v + (u^2 - v) * w^2.
Given g.f. A(x) then B(q) = A(q^3)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) * (1 + u * v) - (2 * u * v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (648 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134132.
G.f.: ( Product_{k>0} (1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2 )^(-1).
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EXAMPLE
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G.f. = 1 - x + x^3 - x^4 - x^5 + 2*x^6 - 2*x^7 + 2*x^9 - 2*x^10 - x^11 + 4*x^12 + ...
G.f. = 1/q - q^5 + q^17 - q^23 - q^29 + 2*q^35 - 2*q^41 + 2*q^53 - 2*q^59 - q^65 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3]^2 QPochhammer[ x^9, x^18] , {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; b := q^(1/6)*eta[q]*eta[q^6]^2* eta[q^9]/(eta[q^2]*eta[q^3]^2*eta[q^18]); a := CoefficientList[ Series[ b, {q, 0, 80}], q]; Table[a[[n]], {n, 1, 80}] (* G. C. Greubel, Jul 03 2018 *)
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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^6 + A)^2 * eta(x^9 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^18 + 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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