%I #25 Dec 15 2024 19:37:23
%S 1,2,2,2,4,8,14,22,36,64,114,198,340,586,1018,1772,3076,5332,9248,
%T 16054,27872,48376,83952,145700,252888,438938,761846,1322286,2295022,
%U 3983384,6913822,12000054,20828006,36150354,62744812,108903838,189020310,328075444,569428264,988335418,1715417004
%N 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.
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.
%H Robert Israel, <a href="/A285636/b285636.txt">Table of n, a(n) for n = 0..600</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F G.f.: A(x) = P(x)/(R(x)*Q(x)), where P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / 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 Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).
%F a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.452642356466453742995961374156022446123012... - _Vaclav Kotesovec_, Aug 26 2017, updated Sep 24 2020
%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 14*x^6 + 22*x^7 + 36*x^8 + 64*x^9 + ...
%p A10:= [1, seq([x^i,1],i=1..10)]: B10:= [1, seq([-x^i,1],i=1..10)]:
%p S:= series(numtheory:-nthconver(A10,10)/numtheory:-nthconver(B10,10),x,51):
%p A:= [seq(coeff(S,x,i),i=0..50)]; # _Robert Israel_, Dec 15 2024
%t nmax = 40; 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 nmax = 40; CoefficientList[Series[Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, 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}] Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}]), {x, 0, nmax}], x]
%Y Cf. A003823, A005169, A007325, A055101, A285635, A285637, A285638.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 23 2017