Does anyone know of a library with a working implementation of backpropagation through time?
Any of Java/Python/C#/VB.NET/F# (preferably the last one) will do!
You can use TensorFlow's dynamic_rnn() function (API doc). TensorFlow's tutorial on Recurrent Neural Networks will help.
Also, this great blog post provides a nice introduction to predicting sequences using TensorFlow. Here's another blog post with some code to predict a time series.