Divisibility of a summation Let $n , l, k, p$ be positive integers, and $1\leq p\leq n$. Let $B(n, l, k, p)$ be the cardinality of the following set
\begin{eqnarray}
\{(a_1, a_2, \cdots, a_n)\in\mathbb{Z}^{\oplus n}|\ \ 0\leq a_i\leq l-1\text{ for }1\leq i\leq n, a_1+\cdots+a_n=lk-p\}.
\end{eqnarray}
Show that 


*

*$\displaystyle\sum_{k=1}^n B(n, l, k, p)=l^{n-1}$, and 

*$l\displaystyle\sum_{k=1}^{2n}((-1)^{k+p}B(2n+1, l, k, p)-(-1)^{k+n}B(2n+1, l, k, n))$ is divisible by $2^{n+1}$, for $1\leq p\leq n$.


I think maybe it is useful to think of the $n$-tuples $(a_1, \cdots, a_n)$ as an $l$-ary number.

For the first question, use the simple fact that for any partial sequence $(a_1, a_2, \ldots, a_{n-1}, ?)$ there is a unique way to fill in the blank such that the sum is congruent to $-p$ mod $l$.  There are $l^{n-1}$ such partial sequences.
