The Infinite K-Tree

Let's define an Infinite K-Tree as follows : - Each vertex has exactly K children - Each edge has some weight - There are Infinite number of nodes in the tree - K edges going out of a vertex has weights 1,2,3, ... K Below is an example of an Infinte 3-Tree (Only 3-Levels are shown) : Find out "How many paths of total weight N (the sum of all weights of the edges in the path) are there, start