## Catalan and Fibonacci Formulae

Using the difference table method described in my previous post, I discovered the following formulae.

The first is not new. I do not know whether the last two are:

$\bigtriangleup_d(n) = \dbinom{n+d-1}{d} = \sum_{k=1}^{d} \dbinom{d-1}{k-1} \dbinom{n}{k}$;

$C_n = \sum_{i=0}^{n} \dbinom{n}{i} \bigtriangleup(F_{i-1})$;

$\sum_{i=0}^{n-1} C_i = \sum_{i=0}^{n} \dbinom{n}{i} \bigtriangleup(F_{i-2})$.

Here, $C_n = \dfrac{1}{n+1}\dbinom{2n}{n}$ is the nth Catalan number, $F_n$ is the nth Fibonacci number, $\bigtriangleup_d(n)$ is the nth d-dimensional tetrahedral number, and $\bigtriangleup(n) = \bigtriangleup_2(n)$ is the nth triangular number.