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A106507
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G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).
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16
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1, -1, 1, -2, 3, -4, 5, -7, 10, -13, 16, -21, 28, -35, 43, -55, 70, -86, 105, -130, 161, -196, 236, -287, 350, -420, 501, -602, 722, -858, 1016, -1206, 1431, -1687, 1981, -2331, 2741, -3206, 3740, -4368, 5096, -5922, 6868, -7967, 9233, -10670, 12306, -14193
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OFFSET
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0,4
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COMMENTS
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REFERENCES
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation.
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LINKS
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FORMULA
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Expansion of 1 / psi(x) in powers of x where psi() is a Ramanujan theta function, which is Jacobi's theta_2(0, sqrt(x))/(2*x^(1/8)) function. See, e.g., the Eric Weisstein link.
Expansion of q^(1/8) * eta(q) / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -1, 1, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4 * (w^4 + 4*v^4) - v^6*w^2.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2*u6^3 + u2^2*u3^3 - u3^3*u6^2.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^3*u6^2 + 3*u1^3*u2^2 - u2^3*u3*u6.
G.f.: Sum_{k>=0} a(k) * x^(8*k - 1) = 1 / (Sum_{k in Z} x^((4k + 1)^2)).
G.f.: 1 / (1 + x + x^3 + x^6 + ...) = 1 - x * (1 - x) / (1 - x^2)^2 + x^4 * (1 - x) * (1 - x^2) / ((1 - x^2)^2 * (1 - x^4)^2) + ... [Ramanujan] - Michael Somos, Jul 21 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A015128. - Michael Somos, Nov 01 2008
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(2*k)). - Michael Somos, Nov 08 2015
G.f.: (x; x^2)_{1/2}, where (a; q)_n is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016
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EXAMPLE
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G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 5*x^6 - 7*x^7 + 10*x^8 + ...
G.f. of B(q) = A(q^8) / q = 1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 + ...
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MATHEMATICA
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nmax=40; CoefficientList[Series[Product[1/(1-x^k)^((-1)^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 28 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/8) / EllipticTheta[ 2, 0, x^(1/2)] , {x, 0, n}]; (* Michael Somos, Jun 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A)^2, n))};
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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