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A260149
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Expansion of f(q) * phi(q) in powers of q where f() is a Ramanujan theta function and phi() is a 6th-order mock theta function.
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0
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1, 0, 2, 0, 2, -2, 0, 0, 2, 0, 2, 0, 0, -4, 2, 2, 2, -4, 0, 0, 2, 2, 2, 0, -2, -4, 4, 0, 2, -4, -2, 0, 2, 2, 4, 0, 0, -4, 0, 2, 4, -4, -2, 0, 2, 0, 0, 0, -2, -4, 6, 2, 2, -4, 0, -2, 2, 4, 4, 0, -2, -4, 0, 0, 2, -6, -2, 0, 2, 4, 2, 0, 0, -4, 4, 0, 2, -2, -2, 0
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OFFSET
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0,3
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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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 4, 5th equation.
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LINKS
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FORMULA
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G.f.: Sum_{k in Z} x^(k*(3*k + 1)/2) * (1 + x^(2*k)) / (1 + x^(3*k)).
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EXAMPLE
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G.f. = 1 + 2*x^2 + 2*x^4 - 2*x^5 + 2*x^8 + 2*x^10 - 4*x^13 + 2*x^14 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 + 2 Sum[ x^(k (3 k + 1)/2) (1 + x^(2 k)) / (1 + x^(3 k)), {k, (Sqrt[ 24 n + 1] - 1) / 6}], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ -x, x] Sum[ (-1)^k x^k^2 QPochhammer[ x, x^2, k] / QPochhammer[ -x, x, 2 k], {k, 0, Sqrt @ n}], {x, 0, n}]];
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, (sqrtint(24*n + 1)-1)\6, 2 * x^(k*(3*k + 1)/2) * (1 + x^(2*k)) / (1 + x^(3*k)), 1 + 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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