What's the formulation of this elements?

Question

I'm trying to find the formulation of these elements.

$N$ is a constant.

$a_k=1$

$a_{k-1} =a_k* \frac{k-1}{N}$

$a_{k-2} = a_{k-1} * \frac{k-2}{N}$

$a_{k-3} =a_{k-2} * \frac{(k-3)}{N}$

.

.

$a_2=\frac{2*a_3}{N}$

$a_1=\frac{a_2}{N}$

By placing the values , I got the formula, but it's not true. Where is my mistake?

For every $ k' : (1<= k' <= k)$

$a_k'=\frac{(k-1)!}{(2k-(k'-1))! *(N^{k-k'})}$

Answer 1

The expression you are looking for is $$ a_j = \frac{(k-1)!}{(j-1)!} N^{j-k} . $$

In fact:

• $a_k=1$,
• $a_j = \frac{j}{N}\frac{(k-1)!}{(j+1-1)!}N^{j+1-k} = \frac{j}{N}a_{j+1}$.