I really need help with the 13-th prob. Chap. 2 in the book Kuratowski "Introduction to Calculus", it says:
We consider the following table (i.e. double sequence $\{a_{n,k}\}$):
\begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & \cdots \\ 1 & 2 & 3 & 4 & 5 & \cdots \\ 1 & 3 & 6 & 10 & 15 & \cdots \\ 1 & 4 & 10 & 20 & 35 & \cdots \\ 1 & 5 & 15 & 35 & 70 & \cdots \\ \cdots & \cdots & \cdots & \cdots &\cdots & \cdots \\ \end{array}
This table is defined as follows: the firs line consists of nothing but 1, i.e. $a_{1,k}=1$; the $k$-th term in the $n$-th line is the sum of the first $k$ terms of the $(n-1)$-th line, i.e. \begin{equation} a_{n,k}=a_{n-1,1}+a_{n-1,2}+ \cdots + a_{n-1,k}. \end{equation} Prove that $$a_{n,k}=\frac{(k+n-2)!}{(k-1)!(n-1)!}$$ and that the terms $a_{n,1}$, $a_{n-1,1}$,..., $a_{1,n}$ are successive coefficients of the Newton expansion of the expression $(a+b)^{n-1}$ (as is easily seen they are the terms of the table lying on the straight line joining the $n$-th term of the first line with the first term of the $n$-th line).
I tried to do induction but i don't know what to do with the two indexes, and when i tried to do something with the formula $a_{n,k}$ it becomes to difficult to manipulate.
I'll appreciate all the help or ideas.
We can work by induction on the quantity $Y=k+n$. If $Y=2$ then $k=1$ and $n=1$ so $$ a_{1, 1}=1=\frac{(1+1-2)!}{(1-1)!(1-1)!} $$
Now suppose that for a certain $Y$ the equality holds for every $k, n$ such that $Y=k+n$, so now set $n, k$ such that $n+k=Y+1$. If $n=1$ then $$ a_{1, k}=\frac{(k-1)!}{(k-1)!}=1 $$
Otherwise if $n>1$ we have $$ a_{n, k}=a_{n-1, 1}+a_{n-1, 2}+\dotsb+a_{n-1, k-1}+a_{n-1, k}=a_{n, k-1}+a_{n-1, k} $$ with $n+k-1=n-1+k=Y$ and so we can use induction to obtain
$$ a_{n, k}=\frac{(k+n-3)!}{(k-2)!(n-1)!}+\frac{(k+n-3)!}{(k-1)!(n-2)!}=\frac{(k+n-3)!}{(k-2)!(n-2)!(n-1)}+\frac{(k+n-3)!}{(k-2)!(n-2)!(k-1)}=\frac{(k+n-3)!(k-1)+(k+n-3)!(n-1)}{(k-2)!(n-2)!(k-1)(n-1)}=\frac{(k+n-3)!(k+n-2)}{(k-1)!(n-1)!}=\frac{(k+n-2)!}{(k-1)!(n-1)!} $$
Obtaining so the initial statement.