Method of expressing the product of first n integers

Question

I am trying to show a pattern whereby the first term is 140 the next term is $140\times139$ and the next $140\times139\times138\dots$ I can do this as follows:

$\frac{140!}{(140-n)!}$ but that doesnt hold for negative numbers i.e. $140\times139\dots\times-1\times\dots$

How can I do this?

Answer 1

$$f(n):=\begin{cases}\tfrac{140!}{(140-n)!}&\text{if }1\le n\le 140,\\ 0&\text{if }n>140.\end{cases} $$

Answer 2

Just use Knuth's notation for falling factorial powers: $$ x^{\underline{k}} = x (x - 1) (x - 2) \ldots (x - k + 1) $$ I.e., $k$ factors, decreasing by 1 each. This is a polynomial of degree $k$ in $x$.

Similarly, raising factorial powers: $$ x^{\overline{k}} = x (x + 1) (x + 2) \ldots (x + k - 1) $$

There are several notations around for those, like $(x)_k$.