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A226556
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Expansion of f(x, -x^4) / f(-x^2, x^3) in powers of x where f(,) is Ramanujan's general theta function.
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2
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1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 2, 3, 2, 0, -2, -4, -4, -3, -1, 3, 6, 7, 5, 0, -5, -9, -10, -7, -1, 7, 14, 16, 11, 1, -11, -20, -22, -16, -2, 15, 29, 33, 23, 2, -23, -41, -45, -32, -4, 30, 57, 64, 45, 4, -43, -78, -86, -60, -7, 57, 107, 119, 83, 8, -79
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OFFSET
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0,11
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COMMENTS
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Given the g.f. A(x), then S(q) := q^(1/5) * A(q) notation is used by Berndt.
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REFERENCES
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G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 57.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 9.
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LINKS
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FORMULA
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Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (1 + u*v^3) - 3 * u^2*v^2.
Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^5 * (1 - 3*v + 4*v^2 - 2*v^3 + v^4) - v * (1 + 2*v + 4*v^2 + 3*v^3 + v^4).
Euler transform of period 20 sequence [1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -1, -1, 0, 1, 0, ...].
G.f.: (Sum_{k in Z} (-1)^k * (-x)^((5*k + 3)*k / 2)) / (Sum_{k in Z} (-1)^k * (-x)^((5*k + 1)*k / 2)).
G.f.: 1 / (1 - x / (1 + x^2 / (1 - x^3 / ...))). [continued fraction]
G.f.: 1/G(0), where G(k)= 1 + (-x)^(k+1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
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EXAMPLE
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G.f. = 1 - x + x^2 - x^4 + x^5 - x^6 + x^7 - x^9 + 2*x^10 - 3*x^11 + 2*x^12 + ...
G.f. = q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + 2*q^61 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ ContinuedFractionK[ (-q)^k, 1, {k, 0, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q^5] QPochhammer[ q^4, -q^5] / (QPochhammer[ q^2, -q^5] QPochhammer[-q^3, -q^5]), {q, 0, n}];
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PROG
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(PARI) {a(n) = my(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + 3*n) / 2), x * O(x^n)) / sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + n) / 2), x * O(x^n)), 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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