Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.
2. From Pascal to Leibniz
In Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025557200004447/resource/name/S0025557200004447_eqn1.gif?pub-status=live)
Specifically, we define the sequence sn; as follows [6]: