Exponentiation of Pascal's Triangle(in matrix form)

Question

I want to find a pattern in subsequent exponentiations of the pascal triangle shown in the form below
Matrix P[K+1][K+1]:
$$ \begin{matrix} \binom{0}{0} & 0 & 0 & 0\cdots &0\\ \binom{1}{0} & \binom{1}{1} & 0 & 0\cdots &0 \\ \binom{2}{0} & \binom{2}{1} & \binom{2}{2} & 0\cdots &0 \\ \binom{3}{0} & \binom{3}{1} & \binom{3}{2} & \binom{3}{3}\cdots &0 \\ \vdots & \vdots &\vdots &\vdots\ddots& \vdots\\ \binom{k}{0} & \binom{k}{1}& \binom{k}{2} & \binom{k}{3}\cdots &\binom{k}{k} \\ \end{matrix} $$ My motive is to compute $P^N$ in $O(KlogN)$ complexity ,which is definately not possible for any ordinary matrix(which would take $O(K^3logN)$).
But I am pretty much sure it is possible for this special matrix as I have done it in $O(K^2logN)$ but the problem(here),which i am solving,suggests it to be done in $O(KlogN)$,Definately there is a secret which I want to find.
Thanks in Advanced.