Definite Integration

For \(m, n > 0\), let \(\alpha(m, n)=\int_\limits{0}^{2} t^{m}(1+3 t)^{n} d t\). If \(11 \alpha(10,6)+18 \alpha(11,5)=p(14)^{6}\), then \(p\) is equal to ___________.