The coefficients of a certain power series

$\displaystyle \displaystyle\mbox{P(s,t)}=\sum_{m=0}^{\infty}\sum _{n=0}^{\infty}a_{m,n}s^mt^n$

satisfy

$\displaystyle 3(m+1)a_{m+1,n}-(n+1)a_{m,n+1}+a_{m,n}=0$

and it is known that $\displaystyle \mbox{P(t,t)}=\exp{(2t)}$. Find $\displaystyle \mbox{P(s,t)}$.

$\displaystyle \displaystyle\mbox{P}_s=\sum_{m=1}^{\infty}\sum_{n =0}^{\infty}a_{m,n}ms^{m-1}t^n$

$\displaystyle \displaystyle\mbox{P}_t=\sum_{m=0}^{\infty}\sum_{n =1}^{\infty}a_{m,n}ns^{m}t^{n-1}$

$\displaystyle \displaystyle c_1\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}a_{m,n}ms ^{m-1}t^n+c_2\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}a_{ m,n}ns^{m}t^{n-1}+c_3\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}a_{m,n }s^mt^n=0$

Is this how it should be approached?