$n$th term of a recursive formula

Question

I have a formula $$ 1 + px + \dfrac{p(p-1)}{1*2}x^2 + \dfrac{p(p-1)(p-2)}{1*2*3} x^3 $$ can someone please tell me what the formula is for the $n$th term of this recursive definition ? Do I have to do it using factorials ?

Answer 1

It is related to the generalized binomial coefficient. The $n$th term is given by:

$$a_n = \frac{1}{n!}\binom{p}{n}x^n$$

Assuming that $p$ is a positive integer and $p \ge n$, we have:

$$a_n = \frac{1}{n!}\frac{p!}{n!(p-n)!} x^n= \frac{p!}{(n!)^2(p-n)!}x^n$$