Denote $$T(m)=\sum\_{1\leq n\_m\leq n\_{m-1}\leq\dots\leq n\_2\leq n\_1\leq m}\prod\_{i=1}^{m}\binom{n\_i}{n\_{i+1}}.$$


Is there a name for this kind of summation and is there a good estimate for $\ln T(m)$ bounded above and below by constant factors (preferably additively)?



Terminology and approximation to logarithm of a sum of products of binomial coefficients
Notice that the product of binomial coefficients can be expressed as a [multinomial coefficient](https://en.wikipedia.org/wiki/Multinomial_theorem):
$$\prod\_{i=1}^m \binom{n\_i}{n\_{i+1}} = \binom{n\_1}{n\_1-n\_2,n\_2-n\_3,\dots,n\_{m-1}-n\_m,n\_m}.$$
Denoting $d\_i:=n\_i-n\_{i+1}$ for $i<m$ and $d\_m:=n\_m$, and noticing that they all are non-negative except $d\_m$ which is positive, we get
$$T(m) = \sum\_{d\_1+d\_2+\dots+d\_m\leq m} \binom{d\_1+d\_2+\dots+d\_m}{d\_1,d\_2,\dots,d\_m} - \sum\_{d\_1+d\_2+\dots+d\_{m-1}\leq m} \binom{d\_1+d\_2+\dots+d\_{m-1}}{d\_1,d\_2,\dots,d\_{m-1}},$$
where the sums are taken over non-negative indices.


Now,
\begin{split}
\sum\_{d\_1+d\_2+\dots+d\_m\leq D} \binom{d\_1+d\_2+\dots+d\_m}{d\_1,d\_2,\dots,d\_m}
&= \sum\_{s=0}^D \sum\_{d\_1+d\_2+\dots+d\_m=s} \binom{s}{d\_1,d\_2,\dots,d\_m}\\
&= \sum\_{s=0}^D m^s \\
&= \frac{m^{D+1}-1}{m-1}.
\end{split}


Therefore,
$$T(m) = \frac{m^{m+1}-1}{m-1} - \frac{(m-1)^{m+1}-1}{m-2}.$$


