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A257628
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Expansion of 1 - f(-x) in powers of x where f() is a Ramanujan theta function.
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3
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0, 1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0
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COMMENTS
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LINKS
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FORMULA
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G.f.: x + x^2 * (1 - x) + x^3 * (1 - x) * (1 - x^2) + ....
G.f.: Sum_{k>0} -(-1)^k * (x^((3*k^2 - k)/2) + x^((3*k^2 + k)/2)).
G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k) / 2) / ((1 - x) * (1 - x^2) * ... * (1 - x^k)).
G.f.: -(Product_{j>=1}(1-x^j) - 1), from Euler's Pentagonal Theorem. - Wolfdieter Lang, Feb 16 2021
a(n) = - A010815(n) unless n=0, a(0) = 0.
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EXAMPLE
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G.f. = x + x^2 - x^5 - x^7 + x^12 + x^15 - x^22 - x^26 + x^35 + x^40 + ...
G.f. = q^25 + q^49 - q^121 - q^169 + q^289 + q^361 - q^529 - q^625 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1 - QPochhammer[ x], {x, 0, n}];
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ n > 0 && IntegerQ[m], - KroneckerSymbol[ 12, m], 0]];
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 1 - eta(x + x * O(x^n)), n))};
(PARI) {a(n) = my(m); if( n>0 && issquare( 24*n + 1, &m), - kronecker( 12, m))};
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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