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A285638 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 A285638 #12 Mar 11 2025 02:54:36
%S A285638 1,0,-1,-2,0,0,-1,-4,-4,-2,-3,-6,-13,-16,-19,-24,-38,-60,-82,-110,
%T A285638 -150,-224,-324,-458,-637,-898,-1289,-1838,-2609,-3680,-5223,-7430,
%U A285638 -10571,-15004,-21272,-30202,-42903,-60960,-86543,-122860,-174450,-247762,-351883,-499668,-709521,-1007532
%N A285638 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 A285638 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A285638 G.f.: A(x) = Q(x)/(R(x)*P(x)), where Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m), 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))) and P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m).
%e A285638 G.f.: A(x) = 1 - x^2 - 2*x^3 - x^6 - 4*x^7 - 4*x^8 - 2*x^9 - 3*x^10 - 6*x^11 - 13*x^12 - ...
%t A285638 nmax = 45; CoefficientList[Series[1/((1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])) (1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))), {x, 0, nmax}], x]
%t A285638 nmax = 45; CoefficientList[Series[Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0,
%t A285638 nmax}] Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y A285638 Cf. A003823, A005169, A007325, A055101, A285635, A285636, A285637.
%K A285638 sign
%O A285638 0,4
%A A285638 _Ilya Gutkovskiy_, Apr 23 2017