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The reason is the following: add a print g[k] after the definition of g[k], you will see the following expressions (for len=10):

-180*x/(180*x - 1) + 1
-144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) + 1
-112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) + 1
-84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) + 1
-60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) + 1
-40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-12*x/(12*x + 1/(24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-4*x/(4*x + 1/(12*x/(12*x + 1/(24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1

So, you can see that they become quite large, so that the computation become very long. Here is a possible explanation.

So, to check our hypothesis, what you have to do is to simplify the expressions to keep something small along the computation. You can work on fraction field, or, if you want to stay in the symbolic ring, then you should replace:

g[k] = 1 - f(k)/(f(k)-1/g[k+1])

with

g[k] = (1 - f(k)/(f(k)-1/g[k+1])).full_simplify()

Now, with len=20, you get very fast:

[1, 4, 64, 2176, 126976, 11321344, 1431568384, 243680935936, 53725527801856, 14893509177769984, 5070334006399074304, 2079588119566033616896, 1011390382859091900891136, 575501120339508919401447424, 378784713733072451034702413824, 285539131625477547496925147693056, 244410463271028625971446390840098816, 235752874763994894181564461756568305664, 254537317950438535955585357041409084882944, 305766504814246648318888059404547150542012416]

The reason is the following: add a print g[k] after the definition of g[k], you will see the following expressions (for len=10):

-180*x/(180*x - 1) + 1
-144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) + 1
-112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) + 1
-84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) + 1
-60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) + 1
-40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-12*x/(12*x + 1/(24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-4*x/(4*x + 1/(12*x/(12*x + 1/(24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1

So, you can see that they become quite large, so that the computation become very long. Here long, which explains why your computation is a possible explanation.slow.

So, to check our hypothesis, hypothesis (and solve the problem), what you have to do is to simplify the expressions to keep something small along the computation. You can work on fraction field, field (very fast), or, if you want to stay in the symbolic ring, then you should replace:

g[k] = 1 - f(k)/(f(k)-1/g[k+1])

with

g[k] = (1 - f(k)/(f(k)-1/g[k+1])).full_simplify()

Now, with len=20, you get very fast:without waiting:

[1, 4, 64, 2176, 126976, 11321344, 1431568384, 243680935936, 53725527801856, 14893509177769984, 5070334006399074304, 2079588119566033616896, 1011390382859091900891136, 575501120339508919401447424, 378784713733072451034702413824, 285539131625477547496925147693056, 244410463271028625971446390840098816, 235752874763994894181564461756568305664, 254537317950438535955585357041409084882944, 305766504814246648318888059404547150542012416]

The reason is the following: add a print g[k] after the definition of g[k], you will see the following expressions (for len=10):

-180*x/(180*x - 1) + 1
-144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) + 1
-112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) + 1
-84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) + 1
-60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) + 1
-40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-12*x/(12*x + 1/(24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1
-4*x/(4*x + 1/(12*x/(12*x + 1/(24*x/(24*x + 1/(40*x/(40*x + 1/(60*x/(60*x + 1/(84*x/(84*x + 1/(112*x/(112*x + 1/(144*x/(144*x + 1/(180*x/(180*x - 1) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) - 1)) + 1

So, you can see that they become quite large, so that the computation become very long, which explains why your computation is slow.

So, to check our hypothesis (and solve the problem), what you have to do is to simplify the expressions to keep something small along the computation. You can work on fraction field (very fast), or, if you want to stay in the symbolic ring, then you should replace:

g[k] = 1 - f(k)/(f(k)-1/g[k+1])

with

g[k] = (1 - f(k)/(f(k)-1/g[k+1])).full_simplify()

Now, with len=20, you get without waiting:

[1,
 4,
 64,
 2176,
 126976,
 11321344,
 1431568384,
 243680935936,
 53725527801856,
 14893509177769984,
 5070334006399074304,
 2079588119566033616896,
 1011390382859091900891136,
 575501120339508919401447424,
 378784713733072451034702413824,
 285539131625477547496925147693056,
 244410463271028625971446390840098816,
 235752874763994894181564461756568305664,
 254537317950438535955585357041409084882944,
 305766504814246648318888059404547150542012416]
 305766504814246648318888059404547150542012416]