This is joint work with Clemens Heuberger and Helmut Prodinger. The paper can be found on the corresponding arXiv-page https://arxiv.org/abs/1612.07286.

The following files contain the computational aspects (like asymptotic expansions) of our results:

- binary-trees.ipynb (Results of Section 2)
- lattice-paths-rdeg.ipynb (Results of Section 3.1)
- lattice-paths-fringe.ipynb (Results of Section 3.2)

If you are not familiar with SageMath-files, then here are some instructions explaining how you can run them.

## Using the SageMath files

This is a short guide on how to use and/or process the ancillary files of this paper. These files contain the computational results of our paper, as well as some numerical comparisons. The computations are carried out with the free open-source mathematics software system SageMath. In order to run our files, you need to access SageMath in some way. In general, there are two viable options for this.

### Option 1: Installing SageMath locally on your computer

In order to install SageMath locally, please follow the instructions in the SageMath installation guide.

As soon as you have a running version of SageMath, you can start a local notebook server with the command

1 |
sage -n jupyter |

Then, a tab in your browser shoud open where you can navigate and select the `ipynb`

-file in order to view its content and to execute our computations.

### Option 2: Using SageMathCloud (online)

Navigate to https://cloud.sagemath.com/ and register for a free account or sign in, if you already have an account. Then, create a new project (or open an existing one) and click the “New”-Button in order to upload the `ipynb`

-file to the cloud. Afterwards, it can be found among the files of your project, and you can open it as usual.

Note that you might need to adapt the kenel to a more recent SageMath-version (everything above `sage-7.4`

should be okay).

## Doctest-Files

In case you have an instance of SageMath up and running on your own computer, you can easily verify whether executing our computations on your computer yields the same results: for the sake of comparison and testing we also provide the following files:

With a working SageMath-instance, the files can be tested by calling

1 |
sage -t file.py |

If the tests were successful the output `All tests passed!`

should appear. By adding the flag `--verbose`

, you can see the tests that are currently running.