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A218173
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Expansion of f(x^7, x^17) - x^2 * f(x, x^23) in powers of x where f(,) is Ramanujan's two-variable theta function.
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1
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1, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 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, 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, -1, 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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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^5, b = x^3.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
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
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Expansion of f(x, x^7) * chi(-x) in powers of x where f(,) is Ramanujan's two-variable theta function and chi() is a Ramanujan theta function.
G.f.: Sum_{k in Z} x^(12*k^2 + 5*k) - x^(12*k^2 + 11*k + 2).
a(n) = -A010815(2*n + 1).
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EXAMPLE
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1 - x^2 - x^3 + x^7 + x^17 - x^25 - x^28 + x^38 + x^58 - x^72 - x^77 + x^93 + ...
q^25 - q^121 - q^169 + q^361 + q^841 - q^1225 - q^1369 + q^1849 + q^2809 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, If[ OddQ[ DivisorSigma[ 0, 48 n + 25]], JacobiSymbol[ 6, Sqrt[48 n + 25]], 0]]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] - QPochhammer[ q]) / 2, {q, 0, 2 n + 1}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] (QPochhammer[ q^2]^3 / QPochhammer[ q]^2/ QPochhammer[ q^4] - 1) / 2, {q, 0, 2 n + 1}]; (* Michael Somos, Nov 09 2014 *)
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PROG
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(PARI) {a(n) = local(m); if( issquare( 48*n + 25, &m), kronecker( 6, m), 0)};
(PARI) {a(n) = local(m); if( n<0, 0, m = 2*n + 1; - polcoeff( eta( x + x * O(x^m)), m))};
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CROSSREFS
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Cf. A010815, A069911.
Sequence in context: A145377 A246260 A275973 * A068426 A267006 A280816
Adjacent sequences: A218170 A218171 A218172 * A218174 A218175 A218176
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Oct 22 2012
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STATUS
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approved
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