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Search: id:A134131
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| A134131 |
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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. |
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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, -191, -38, 251, -217, -62, 320, -279, -57, 368, -318, -88, 464
(list; graph; listen)
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OFFSET
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0,7
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FORMULA
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Expansion of chi(-q) * chi(-q^9) / chi(-q^3)^2 in power of q where chi() is a Ramanujan theta function.
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(x) = A(x^6) / x satisfies 0 = f(B(x), B(x^2), B(x^4) ) where f(u, v, w) = (u^2 + v) * v + (u^2 - v) * w^2.
Given g.f. A(x) then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^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)) = 1 / f(t) where q = exp(2 pi i t).
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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1/q - q^5 + q^17 - q^23 - q^29 + 2*q^35 - 2*q^41 + 2*q^53 - 2*q^59 - ...
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PROGRAM
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(PARI) {a(n) = local(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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A112178(3*n) = a(n). Convolution inverse of A134132.
Sequence in context: A111165 A029321 A029310 this_sequence A127527 A141661 A080769
Adjacent sequences: A134128 A134129 A134130 this_sequence A134132 A134133 A134134
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Oct 10 2007
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