This is equivalent to the number of paths from $(0,0)$ to $(n,n)$ allowing -any- lattice steps that move closer to $(n,n)$ (this takes into account 'repeated points in the chain). Another way of saying this is integer compositions of $(n,n)$ (adding any sequence of pairs from $(i,j)$, where $i,j$ can run independently from 0 to $n$, but at least one is greater than 0).
The recurrence for $F(n,m,k)$ is $ F(m,n,k) = \sum_{i=0}^{n-1} \sum_{j=0}^{m-1} F(i,j,k-1) + \sum_{i=0}^{n-1} F(i,m,k-1)+ \sum_{j=0}^{m-1} F(n,j,k-1) $ which converts to
$ \begin{eqnarray} F(0,0,0) &=& 1\\ F(n,m,0) &=& 0\\ F(n,m,k) &=& F(n,m-1,k) + F(n-1,m,k) - F(n-1, m-1,k) +\\ && F(n,m-1,k-1) + F(n-1,m,k-1) - F(n-1, m-1,k-1) \end{eqnarray} $
The following table is for $\sum_{k\ge0} F(n,m,k)$:
1 1 2 4 8 16 32 64 128 256 1 3 8 20 48 112 256 576 1280 2 8 26 76 208 544 1376 3392 4 20 76 252 768 2208 6080 8 48 208 768 2568 8016 16 112 544 2208 8016 32 256 1376 6080 64 576 3392 128 1280 256
with main diagonal: $ 1, 3, 26, 252, 2568, 26928, 287648, 3112896,... $
the number of paths to $(n,n)$ of any length (1 to 2n), which is OEIS sequence A02141, closely related to the central Delannoy numbers({OEIS A001850](http://oeis.org/A001850)).
The original question, for the table $F(n,n,k)$ for those paths of length $k$, this is given by OEIS A059576.