What is the generating function of $f_{m,n}$?

$ f_{m,n} = \begin{cases} 0 , & \text{if $m<0 $ or $ n<0$ }; \\ f_{n,m} , & \text{ if $n<m$}; \\ 1, & \text{ if $0=m$ and $ n\in\{0,1\} $}; \\ f_{0 ,n-1}+ f_{1,n-1}, & \text{ if $0=m$ and $ n>1 $}; \\ f_{m-1 ,n}+ f_{m ,n-1}+ f_{m-1,n-1}, & \text{ if $0<m\in \{n,n-1\} $}; \\ f_{m-1 ,n}+ f_{ m ,n-1}+ f_{m-1,n-1} + f_{m+1,n-1},& \text{ if $0<m<n-1$}. \end{cases} $