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 A106507 G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)). 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present entry gives 1/psi(q). For various G.f. versions see the reciprocals of the ones given in A010054. - Wolfdieter Lang, Jul 05 2016 REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 C. Adiga, N. Anitha, T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, arXiv:math/0501528 [math.NT], 2005. See page 5. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol FORMULA 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 a(n) = (-1)^n * A006950(n). Convolution inverse of A010054. Series reversion of A106336. - Michael Somos, May 10 2012 a(2*n) = A233758(n). a(2*n + 1) = - A233759(n). - Michael Somos, Nov 05 2015 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 a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 08 2017 EXAMPLE 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 + ... MATHEMATICA 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 *) (QPochhammer[x, x^2, 1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A)^2, n))}; CROSSREFS Cf. A006950, A010054, A106336, A233758, A233759. Sequence in context: A316720 A316721 A316722 * A006950 A052335 A193771 Adjacent sequences:  A106504 A106505 A106506 * A106508 A106509 A106510 KEYWORD sign,easy AUTHOR Michael Somos, May 04 2005 EXTENSIONS Definition changed by N. J. A. Sloane, Aug 14 2007 STATUS approved

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)