Can this recurrence be simplified?

Question

Let $A$ be the set of elements $\{0,1,2,\dots,n-1\}$ and $B_i=\{T(i,1),T(i,2),\dots,T(i,k)\}$ be the subset of $A$, which represent $i$-th combination without repetition of elements of the $A$ (also $0\leqslant T(i,1)<T(i,2)<\dots<T(i,k)<n$) . Then we have for $0<j\leqslant k$ $$T(i,j)=[i=1](j-1)+[1<i\leqslant \binom{n}{k}](([j=k]+[j<k][C_{j+1}])([C_j](a_1-a_2)+a_2)+[j<k](1-[C_{j+1}])(a_2-1))$$ where $$[C_j]=[T(i-1,j)=n-k+j-1]$$ and $$a_1=T(i,j-1)+1, a_2=T(i-1,j)+1$$ So can this recurrence be simplified? Is there (which looks impossible) a closed form for $T(i,j)$?