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A259920
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Expansion of phi(-x^5) * f(-x^5) / f(-x, -x^4) in powers of x where phi() and f() are Ramanujan theta functions.
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1
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1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 5, 3, 5, 2, 6, 3, 6, 3, 7, 4, 7, 5, 9, 5, 9, 5, 11, 6, 11, 7, 14, 7, 15, 9, 17, 9, 17, 9, 21, 11, 21, 12, 25, 13, 25, 15, 29, 16, 31, 17, 35, 19, 37, 21, 42, 22, 44, 25, 49, 27, 52, 29, 58, 32, 61
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OFFSET
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0,5
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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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 8th equation.
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LINKS
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FORMULA
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Expansion of f(-x^5)^3 / (f(-x^10) * f(-x^2, -x^3)) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of phi(-x^5) * G(x) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 10 sequence [ 1, 0, 0, 1, -2, 1, 0, 0, 1, -1, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^(5*k^2)) / (Product_{k in Z} 1 - x^abs(5*k + 1)).
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EXAMPLE
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G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^-1 + q^59 + q^119 + q^179 + 2*q^239 + q^359 + q^419 + 2*q^479 + q^539 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 0, -1, 2, -1, 0, 0, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [1, -1, 0, 0, -1, 2, -1, 0, 0, -1][k%10+1]), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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