Given a recurrence relation $f(x, y) = 2 \times f(x-1, y) + 4\times f(x - 1, y - 1)$ for $x, y \geq 1$, otherwise $f(x,y) = 1$. Find the closed-form solution of $f(x,y)$ for all $x$ and $y$.
I tried to calculate a few values, but I can't recognize a pattern from these values, apart from $f(x,y) = 6^x$ for $1 \leq x \leq y$ $$\begin{array}{c|cc} x\backslash y&0&1&2&3\\ \hline 0&1&1&1&1\\ 1&1&6&6&6\\\ 2&1&16&36&36\\ 3&1&36&136&216\\ \end{array}$$
I read several questions similar from this but it didn't fit my question well. Can someone lend me a help on this?