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A113430
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Expansion of f(-x) * f(-x^10) / f(-x^2, -x^8) in powers of x where f() is Ramanujan's two variable theta function.
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2
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1, -1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 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, 0, 0, 0, -1, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^4, b = x.
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
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity
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FORMULA
| Expansion of f(x^7, x^8) - x * f(x^2, x^13) in power of x.
Expansion of G(x^2) * f(-x) where G() is g.f. of A003114.
Euler transform of period 10 sequence [ -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, ...].
|a(n)| is the characteristic function of the numbers in A093722.
The exponents in the q-series q * A(q^120) are the square of the numbers in A057538.
G.f.: Prod_{k>0} (1 - x^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))) = Sum_{k} x^((15*k^2 + k) / 2) - x^((15*k^2 - 11*k + 2) / 2).
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EXAMPLE
| 1 - x - x^3 + x^7 + x^8 - x^14 - x^20 + x^29 + x^31 - x^42 - x^52 + ...
q - q^121 - q^361 + q^841 + q^961 - q^1681 - q^2401 + q^3481 + q^3721 + ...
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PROG
| (PARI) {a(n) = local(m); if( n<0 || !issquare( n*120 + 1, &m) || 1!=gcd(m, 30), 0, (-1)^(m%30\10))}
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k * [1, 1, 0, 1, 1, 1, 1, 1, 0, 1][k%10 + 1], 1 + x * O(x^n)), n))}
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CROSSREFS
| Cf. A003114, A010815, A057538, A093722.
Sequence in context: A181101 A143259 * A113681 A179416 A155972 A010054
Adjacent sequences: A113427 A113428 A113429 * A113431 A113432 A113433
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 31 2005
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