%I #50 Mar 12 2021 22:24:43
%S 1,-1,1,-2,3,-4,5,-7,10,-13,16,-21,28,-35,43,-55,70,-86,105,-130,161,
%T -196,236,-287,350,-420,501,-602,722,-858,1016,-1206,1431,-1687,1981,
%U -2331,2741,-3206,3740,-4368,5096,-5922,6868,-7967,9233,-10670,12306,-14193
%N G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present entry gives 1/psi(q).
%C For various G.f. versions see the reciprocals of the ones given in A010054. - _Wolfdieter Lang_, Jul 05 2016
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation.
%H Seiichi Manyama, <a href="/A106507/b106507.txt">Table of n, a(n) for n = 0..10000</a>
%H C. Adiga, N. Anitha, T. Kim, <a href="https://arxiv.org/abs/math/0501528">Transformations of Ramanujan's Summation Formula and its Applications</a>, arXiv:math/0501528 [math.NT], 2005. See page 5.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F 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.
%F Expansion of q^(1/8) * eta(q) / eta(q^2)^2 in powers of q.
%F Euler transform of period 2 sequence [ -1, 1, ...].
%F 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.
%F 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.
%F 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.
%F G.f.: Sum_{k>=0} a(k) * x^(8*k - 1) = 1 / (Sum_{k in Z} x^((4k + 1)^2)).
%F 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
%F 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
%F a(n) = (-1)^n * A006950(n). Convolution inverse of A010054.
%F Series reversion of A106336. - _Michael Somos_, May 10 2012
%F a(2*n) = A233758(n). a(2*n + 1) = - A233759(n). - _Michael Somos_, Nov 05 2015
%F G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(2*k)). - _Michael Somos_, Nov 08 2015
%F G.f.: (x; x^2)_{1/2}, where (a; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 20 2016
%F a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 08 2017
%e G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 5*x^6 - 7*x^7 + 10*x^8 + ...
%e 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 + ...
%t nmax=40; CoefficientList[Series[Product[1/(1-x^k)^((-1)^k),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, May 28 2015 *)
%t a[ n_] := SeriesCoefficient[ 2 x^(1/8) / EllipticTheta[ 2, 0, x^(1/2)] , {x, 0, n}]; (* _Michael Somos_, Jun 25 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^2], {x, 0, n}]; (* _Michael Somos_, Nov 08 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] / QPochhammer[ x^4], {x, 0, n}]; (* _Michael Somos_, Nov 08 2015 *)
%t (QPochhammer[x, x^2, 1/2] + O[x]^50)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A)^2, n))};
%Y Cf. A006950, A010054, A106336, A233758, A233759.
%K sign,easy
%O 0,4
%A _Michael Somos_, May 04 2005
%E Definition changed by _N. J. A. Sloane_, Aug 14 2007