How would I go about calculating $D(a,b)$ the number of such paths for some a,b. Say $a,b<=4$ and then express $D(a,b)$ in terms of another delannoy number?
I have calculated $D(a,b)$ using a recurrence relation without any restrictions but dont know what to do with the $y=x$ line.
These numbers are given by the Schröder triangle, OEIS A$106579$, with indexing offset by $1$: $T(n,k)$ in that entry is the number of Delannoy paths from $\langle 0,0\rangle$ to $\langle n-1,k-1\rangle$ that do not drop below the diagonal. The most useful link given seems to be the one to E. Pergola and R. A. Sulanke, Schröder triangles, paths, and parallelogram polyominoes, J. Integer Seq., Article $98.1.7.$, $\mathbf{1}$ ($1998$). That paper uses the notation $R(m,n)$ for the number of Delannoy paths from $\langle 0,0\rangle$ to $\langle m,n\rangle$ that do not drop below the diagonal.
In Section $2$ the authors give the recurrences
$$R(m,n)=R(m,n-1)+2\sum_{j\ge 1}R(m-j,n-1)\;,$$
with initial conditions $R(0,0)=1$, and $R(m,n)=0$ if $m<0$ or $n<m$, and, more usefully,
$$R(m,n)=R(m,n-1)+R(m-1,n-1)+R(m-1,n)$$
with the same initial conditions. In Section $6.1$ they derive the formula
$$R(m,n)=\frac{n-m+1}{n+1}\sum_{k=0}^m\binom{n+1}{m-k}\binom{n+k}k\;.$$