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A108494
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Expansion of f(-q) / f(q) in powers of q where f() is a Ramanujan theta function.
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5
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1, -2, 2, -4, 6, -8, 12, -16, 22, -30, 40, -52, 68, -88, 112, -144, 182, -228, 286, -356, 440, -544, 668, -816, 996, -1210, 1464, -1768, 2128, -2552, 3056, -3648, 4342, -5160, 6116, -7232, 8538, -10056, 11820, -13872, 16248, -18996, 22176, -25844, 30068, -34936, 40528
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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T. Miwa, Integrable Lattice Models and Branching Coefficients, Proceedings of the International Congress of Mathematicians, Vol. 1, (Berkeley, Calif., 1986), 862-870, Amer. Math. Soc., Providence, RI, 1987. MR0934288 (89h:82051)
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LINKS
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FORMULA
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Expansion of (1 - k^2)^(1/8) = k'^(1/4) in powers of q = exp(-Pi K'/K).
Expansion of (theta_4(q) / theta_3(q))^(1/2) = (phi(-q) / phi(q))^(1/2) = chi(-q) / chi(q) = psi(-q) / psi(q) = f(-q) / f(q) where phi(), chi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q)^2 eta(q^4) / eta(q^2)^3 in powers of q.
Euler transform of period 4 sequence [ -2, 1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = -2*u^2 + v^4 + u^4*v^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^4 + 2*u*v -2*u^3*v^3 - v^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -2*u + v*w^2 + u^2*v*w^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2*u2*u6 - 2*u1*u3 + u2*u3^2*u6.
G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = 4 * u*v *(1 - u^4) * (1 + v^4) - (v^2 - u^2) * (u + v)^4. - Michael Somos, Sep 11 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 + x^(2*k - 1)).
Empirical: sum(exp(-Pi)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = 2^(1/8). - Simon Plouffe, Feb 20 2011
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(7/8)/2. - Simon Plouffe, Feb 20 2011
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 23 2017
G.f.: exp(-2*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019
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EXAMPLE
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1 - 2*q + 2*q^2 - 4*q^3 + 6*q^4 - 8*q^5 + 12*q^6 - 16*q^7 + 22*q^8 - ...
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MATHEMATICA
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PROG
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(PARI) {a(n) = if(n<0, 0, polcoeff( prod(k=1, (n+1)\2, (1 - x^(2*k -1)) / (1 + x^(2*k - 1)), 1 + x * O(x^n)), n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3, n))}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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