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.

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