Closed form for the order of the DCell network

Determine a closed-form expression for the order t_{k,n} of the DCell network D_{k,n}, which is recursively defined by t_{0,n}=n and t_{k,n}=t_{k-1,n}(t_{k-1,n}+1).

Background

The DCell network D_{k,n} has order t_{k,n}, where the sequence t_{k,n} is defined recursively. Although the paper cites bounds and an asymptotic representation showing that the order is doubly exponential, it states that no closed form for t_{k,n} is known. This problem concerns obtaining an explicit nonrecursive formula for the network order.