TITLE: Asymptotic Behavior of Iterated Sums
QUESTION [7 upvotes]: Given the integral identity
\begin{align}
\int_{0}^{t} \cdots \int_{0}^{t - t_{1} - \dots - t_{n -1}} 1 \ dt_1 \cdots dt_n = \frac{t^{n}}{n!},
\end{align}
I believe it is true that 
\begin{align}
\sum_{i_{1} = 0}^{\lfloor t \rfloor} \cdots \sum_{i_{n} = 0}^{\lfloor t - i_1 - \cdots - i_{n-1} \rfloor} 1 = \frac{t^{n}}{n!} + O(t^{n-1}),
\end{align}
where $\lfloor \cdot \rfloor$ is the floor function. However, I'm having difficulty in justifying the exponent $n - 1$. Is it indeed the case? Any help would certainly be appreciated.
Thanks!

REPLY [9 votes]: The sum is equal to $\binom{\lfloor t \rfloor + n}{n}$ (choose the distinct numbers $i_1+1, i_1+i_2+2, \ldots , i_1 + \ldots + i_n + n$ in $[ 1 , \lfloor t \rfloor + n ]$), so it is a polynomial in $\lfloor t \rfloor$ of degree $n$ and leading coefficient $1/n!$, and the asymptotic behavior you guessed is correct.