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A267852
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Expansion of psi(x) * psi(x^9) * f(-x^3) / psi(x^3)^2 in powers of x where psi(), and f() are Ramanujan theta functions.
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1
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1, 1, 0, -2, -3, 0, 2, 4, 0, -5, -5, 0, 9, 8, 0, -12, -14, 0, 16, 20, 0, -23, -25, 0, 36, 37, 0, -47, -54, 0, 60, 71, 0, -84, -91, 0, 115, 121, 0, -149, -164, 0, 188, 210, 0, -245, -264, 0, 321, 343, 0, -406, -443, 0, 505, 554, 0, -641, -687, 0, 813, 863, 0
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OFFSET
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0,4
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COMMENTS
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REFERENCES
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13, 10th equation.
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LINKS
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FORMULA
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Expansion of f(-x) * f(-x^6) * f(-x^3, -x^15) / f(-x, -x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of q^(-5/8) * eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^2 / (eta(q) * eta(q^6)^4 * eta(q^9)) in powers of q.
Euler transform of period 18 sequence [ 1, -1, -2, -1, 1, 0, 1, -1, -1, -1, 1, 0, 1, -1, -2, -1, 1, -1, ...].
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EXAMPLE
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G.f. = 1 + x - 2*x^3 - 3*x^4 + 2*x^6 + 4*x^7 - 5*x^9 - 5*x^10 + 9*x^12 + ...
G.f. = q^5 + q^13 - 2*q^29 - 3*q^37 + 2*q^53 + 4*q^61 - 5*q^77 - 5*q^85 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ x^(-1/2) QPochhammer[ x^3] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(3/2)]^2, {x, 0, n}];
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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^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^2 / (eta(x + A) * eta(x^6 + A)^4 * eta(x^9 + 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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