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A208589
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Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
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2
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1, 2, 0, 0, 1, -2, 0, 0, -1, 4, 0, 0, 0, -6, 0, 0, 1, 8, 0, 0, 0, -12, 0, 0, -1, 18, 0, 0, -1, -24, 0, 0, 2, 32, 0, 0, 1, -44, 0, 0, -2, 58, 0, 0, -1, -76, 0, 0, 2, 100, 0, 0, 1, -128, 0, 0, -3, 164, 0, 0, -1, -210, 0, 0, 4, 264, 0, 0, 2, -332, 0, 0, -5, 416
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OFFSET
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0,2
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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).
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LINKS
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Table of n, a(n) for n=0..73.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of q^(1/2) * eta(x^2)^5 / (eta(x)^2 * eta(x^4) * eta(x^8)^2) in powers of q.
Given g.f. A(x), then B(x) = (A(x^2) / x)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v^2 - (v - 4) * (u^2 - 8*u + 16).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = 2 * A083365(n).
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EXAMPLE
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1 + 2*x + x^4 - 2*x^5 - x^8 + 4*x^9 - 6*x^13 + x^16 + 8*x^17 - 12*x^21 + ...
1/q + 2*q + q^7 - 2*q^9 - q^15 + 4*q^17 - 6*q^25 + q^31 + 8*q^33 + ...
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2), n))}
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CROSSREFS
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Cf. A029838, A083365.
Sequence in context: A025875 A026840 A025873 * A112171 A112172 A093085
Adjacent sequences: A208586 A208587 A208588 * A208590 A208591 A208592
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Feb 29 2012
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STATUS
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approved
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