[ Pascal's Triangle and Newton's Binomial ]

Calculate the generating functions of the following sequences:
a) $a_{n}=C_{m+n}^{m}$
b) $a_{n}=C_{n}^{m}$

#
The first relation is a disguised case of the binomial theorem, and the second one is obtained from it by multiplying by $x^{m}$.

## Solution

a) Since $C_{m+n}^{m}=C_{m+n}^{n}=\frac{(m+1)(m+2) \ldots(m+n)}{n!}=(-1)^{n} C_{-(m+1)}^{n}$, then

$$
\sum_{n=0}^{\infty} C_{m+n}^{n} x^{n}=\frac{1}{(1-x)^{m+1}}
$$

b) This function is obtained from the previous one by multiplying by $x^{m}$.

## Answer

a) $\frac{1}{(1-x)^{m+1}}$
b) $\frac{x^{m}}{(1-x)^{m+1}}$.