On a conjecture of Chowla et al (Q1273203)

Let \[ N(m,n)= \sum_{k=0}^n \binom {n}{k}^2 \binom{n+k}{k}^m \] where \(m\geq 0\). The author proves several congruences involving the \(N(m,n)\), for example: \[ N(m,p)\equiv 1+2^m\pmod{p^3} \] if the prime \(p>3\).