IR-TST 2013 Problem 3

For nonnegative integers $m$ and $n$ , define the sequence $a(m,n)$ of real numbers as follows. Set $a(0,0)=2$ and for every natural number $n$ , set $a(0,n)=1$ and $a(n,0)=2$ . Then for $m,n\geq1$ , define $$a(m,n)=a(m-1,n)+a(m,n-1).$$ Prove that for every natural number $k$ , all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.