Question 98
For nonnegative integers $m$ and $n$, the sequence $a(m, n)$ of real numbers is defined as follows: $a(0, 0)$ is equal to $2$, and for each natural number $n$, $a(0, n) = 1$ and $a(n, 0) = 2$. Also for $m, n \in \mathbb{N}$:
$$
a(m, n) = a(m - 1, n) + a(m, n - 1)
$$
Prove that for each natural number $k$, all roots of the polynomial $P_k(x) = \sum_{i=0}^{k} a(i, 2k + 1 - 2i)x^i$ are real numbers.
$$
a(m, n) = a(m - 1, n) + a(m, n - 1)
$$
Prove that for each natural number $k$, all roots of the polynomial $P_k(x) = \sum_{i=0}^{k} a(i, 2k + 1 - 2i)x^i$ are real numbers.