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A204220
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Expansion of f(-x^2, -x^3) * f(-x^2) / f(-x) in powers of x.
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2
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1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 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, 0, 0
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OFFSET
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0,1
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COMMENTS
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f(a, b) := Sum_{k} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's two-variable theta function and f(-x) := f(-x, -x^2).
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LINKS
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Table of n, a(n) for n=0..85.
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FORMULA
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Expansion of G(x) * f(-x^2) where G() is g.f. of A003114.
Expansion of f(-x^13, -x^17) + x * f(-x^7, -x^23) in powers of x.
Euler transform of period 10 sequence [ 1, -1, 0, 0, 0, 0, 0, -1, 1, -1, ...].
G.f.: Sum_{k} (-1)^k * x^(15*k^2) * (x^(2*k) + x^(8*k + 1)) = Product_{k>0} (1 - x^(10*k)) * (1 - x^(10*k -2)) * (1 - x^(10*k -8)) / ((1 - x^(10*k -1)) * (1 - x^(10*k -9))).
|a(n)| is the characteristic function of A204221.
The exponents in the q-series q * A(q^15) is the square of the numbers == +- 1 or +- 4 (mod 15).
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EXAMPLE
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1 + x - x^8 - x^13 - x^17 - x^24 + x^45 + x^56 + x^64 + x^77 - x^112 + ...
q + q^16 - q^121 - q^196 - q^256 - q^361 + q^676 + q^841 + q^961 + q^1156 + ...
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k) ^ ([1, -1, 1, 0, 0, 0, 0, 0, 1, -1][k%10 + 1]), 1 + x * O(x^n)), n))}
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CROSSREFS
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Cf. A003114, A010815.
Sequence in context: A016373 A167700 A010057 * A214263 A016355 A016402
Adjacent sequences: A204217 A204218 A204219 * A204221 A204222 A204223
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jan 13 2012
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STATUS
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approved
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