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A121373
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Expansion of f(x) = f(x, -x^2) in powers of x where f(,) is the Ramanujan two-variable theta function.
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757
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1, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 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^3, b = -x. - Michael Somos, Jul 11 2012
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LINKS
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Table of n, a(n) for n=0..104.
Guo-Niu Han, Enumeration of Standard Puzzles
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 q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4)) in powers of q.
Euler transform of period 4 sequence [ 1, -2, 1, -1, ...].
a(n) = b(24*n + 1) where b(n) is multiplicative and b(p^2e) = (-1)^e if p == 7, 11, 13, 17 (mod 24), b(p^2e) = +1 if p == 1, 5, 19, 23 (mod 24) and b(p^(2e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f.: (1 + x) * (1 - x^2) * (1 + x^3) * (1 - x^4) ...
G.f.: 1 + x - x^2*(1 + x) + x^3*(1 + x)*(1 - x^2) - x^4*(1 + x)*(1 - x^2)*(1 + x^3) + ...
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = a(n).
G.f. Sum_{k>=0} a(k) x^(24*k + 1) = Sum_{k} (-1)^[(k+1)/2] x^(6*k + 1)^2.
a(n) = (-1)^n * A010815(n). |a(n)| = A080995(n).
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EXAMPLE
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1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 + ...
q + q^25 - q^49 - q^121 - q^169 - q^289 + q^361 + q^529 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k, {k, n}], {x, 0, n}] (* Michael Somos, Nov 14 2011 *)
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PROG
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(PARI) {a(n) = if( issquare(24*n + 1, &n), kronecker(24, n))}
(PARI) {a(n) = if( n<0, 0, polcoeff( eta( -x + x * O(x^n)), n))}
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CROSSREFS
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Cf. A010815, A080995.
Sequence in context: * A133985 A143062 A199918 A074910 A115356 A115359
Adjacent sequences: A121370 A121371 A121372 * A121374 A121375 A121376
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jul 24 2006
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STATUS
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approved
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