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A138559
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Expansion of q^(1/24) * eta(q^2)^4 / (eta(q^4)^2 * eta(q)) in powers of q.
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1
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1, 1, -2, -1, 1, -1, -1, -1, 2, 2, -2, 0, 1, 1, -1, 0, 3, 1, -3, -2, 3, 0, -2, -1, 3, 2, -4, -2, 2, 1, -4, -2, 5, 3, -6, -1, 5, 1, -5, -3, 6, 3, -6, -3, 7, 2, -6, -2, 9, 5, -10, -5, 9, 3, -9, -4, 11, 6, -12, -4, 11, 5, -12, -5, 14, 6, -16, -7, 15, 5, -16, -7, 19, 9, -20, -8, 19, 7, -20, -10, 24, 11, -25, -11, 24, 9, -26, -11, 29, 13, -31, -13
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OFFSET
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0,3
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
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Table of n, a(n) for n=0..91.
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 phi(q) * chi(-q) = phi(-q^2) * chi(q) = psi(q) * chi(-q^2)^2 = f(-q) * chi(q)^2 = f(q) * chi(-q^2) = f(q)^2 / psi(q) = phi(-q^2)^2 / f(-q) where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 1, -3, 1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A096920.
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2.
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EXAMPLE
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1/q + q^23 - 2*q^47 - q^71 + q^95 - q^119 - q^143 - q^167 + 2*q^191 + ...
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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)^4 / eta(x^4 + A)^2 / eta(x + A), n))}
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CROSSREFS
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Sequence in context: A118383 A115766 A108339 * A073454 A124765 A080356
Adjacent sequences: A138556 A138557 A138558 * A138560 A138561 A138562
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 24 2008
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STATUS
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approved
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