Chaining recursive functions

Question

$T(0,r) = k$

$T(n,r) = k + T(n-1,r) + T(n-1,r+1)$

1. $T(n,0) = k + T(n-1,0) + T(n-1,1)$
2. $T(n,1) = k + T(n-1,1) + T(n-1,2)$
3. $T(n,2) = k + T(n-1,2)$
4. $T(n,3) = k$

Compact: $T(n,0) = k' + T(n-1,0) + T(n-2,1) + T(n-3,2)$

Is this correctly chained?