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A285635 G.f.: (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) / (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction.

%I A285635 #11 Mar 10 2025 06:47:22
%S A285635 1,-2,2,-2,0,0,-2,2,-4,0,-2,-6,0,-10,-6,-12,-20,-20,-40,-46,-68,-104,
%T A285635 -132,-204,-280,-394,-578,-790,-1154,-1616,-2294,-3286,-4614,-6610,
%U A285635 -9340,-13278,-18878,-26748,-38060,-53978,-76684,-108912,-154600,-219622,-311812,-442818,-628866,-892962,-1268168
%N A285635 G.f.: (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) / (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction.
%H A285635 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A285635 G.f.: A(x) = R(x)*Q(x)/P(x), where R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3))), Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m) and P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m).
%e A285635 G.f.: A(x) = 1 - 2*x + 2*x^2 - 2*x^3 - 2*x^6 + 2*x^7 - 4*x^8 - 2*x^10 - 6*x^11 - ...
%t A285635 nmax = 48; CoefficientList[Series[(1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))/(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
%t A285635 nmax = 48; CoefficientList[Series[Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}] Sum[(-1)^k x^(k^2)/Product[(1 -
%t A285635 x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A285635 Cf. A003823, A005169, A007325, A055101, A285636, A285637, A285638.
%K A285635 sign
%O A285635 0,2
%A A285635 _Ilya Gutkovskiy_, Apr 23 2017