I've written a function. I believe the function runs in complexity:
$$ T(m, n) = \begin{cases} T(m-1, n) + T(m, n-1) + T(m-1, n-1) + 1 &\text{if $m,n>0$}\\ 0 &\text{otherwise} \end{cases} $$
I have no idea how to simplify this. Can anyone help?
(No, this is not a homework assignment. Well, writing the function was, but after writing it I'm certain that I wrote a very inefficient implementation and am trying to figure out just how bad my version really is.)
Let $P = \sum_{n,m \geq 0} T(n,m) x^n y^m$. We have $$ \begin{align*} P &= \sum_{n,m \geq 1} T(n,m) x^n y^m \\ &= \sum_{n,m \geq 1} [T(n-1,m) + T(n,m-1) + T(n-1,m-1)+1] x^n y^m \\ &= \sum_{n \geq 0, m \geq 1} T(n,m) x^{n+1} y^m + \sum_{n \geq 1, m \geq 0} T(n,m) x^n y^{m+1} + \sum_{n,m \geq 0} T(n,m) x^{n+1} y^{m+1} + \sum_{n,m \geq 1} x^n y^m \\ &= (x+y+xy)P + \frac{xy}{(1-x)(1-y)}. \end{align*} $$ We deduce that $(1-x-y-xy)P = xy/(1-x)(1-y)$, and so $$ P = \frac{xy}{(1-x)(1-y)(1-x-y-xy)}. $$
At this point we could invoke multivariate asymptotic analysis, but instead we'll connect $P$ to the Delannoy numbers $D(n,m)$, which follow a similar recurrence $D(n,m) = D(n-1,m) + D(n,m-1) + D(n-1,m-1)$, with base case $D(0,0) = 1$. These numbers have generating series $Q = 1/(1-x-y-xy)$. Our formula above shows that $$ P = \frac{x}{1-x} \frac{y}{1-y} Q = \sum_{n,m \geq 1} x^n y^m Q, $$ and so we obtain the formula $$ T(n,m) = \sum_{n'<n,m'<m} D(n',m'). $$ This formula can also be derived directly in various ways.
Pemantle and Wilson shows that when $n,m \to \infty$ and $n/m,m/n$ are bounded, $$ D(n,m) \sim \frac{nm}{\sqrt{2\pi}(n+m-\sqrt{n^2+m^2})^2\sqrt{n^2+m^2}} \left(\frac{m}{\sqrt{n^2+m^2}-n}\right)^m \left(\frac{n}{\sqrt{n^2+m^2}-m}\right)^n $$ This should be compared to the similar frightening formula for the binomial coefficients, $$ \binom{n+m}{m} \sim \sqrt{\frac{n+m}{2\pi nm}} \left(\frac{n+m}{n}\right)^n \left(\frac{n+m}{m}\right)^m. $$ The asymptotic formula for $D(n,m)$ can probably be used to come up with an asymptotic formula for $T(n,m)$, or one can do the asymptotic analysis directly using the same methods, but I doubt that it would be very useful.
Hint: expand the equation: $$T(m,n) = T(m-2,n) + T(m-2, n-1) + T(m,n-2) + T(m-1,n-2) + 3T(m-1,n-1) + 1$$
As $T(m,n)$ is related to $3T(m-1,n-1)$ we can say $T(m,n) = \Omega(3^{min(m,n)})$.
