The exercise states:
Let $R$ be a (positive-dimensional, finitely generated) homogeneous algebra over a field $\kappa$. Then there exist integers $a_1\ge a_2 \ge \cdots \ge a_j$, such that \begin{align} P_R(n) = {n+a_1 \choose a_1}+{n+a_2-1 \choose a_2}+\cdots+{n+a_j-(j-1) \choose a_j}, \end{align} where $P_R(n)$ is the Hilbert polynomial of $R$.
By Corollary 4.2.14, we have that $H(R,n) = H(R,n-1)^{\langle n-1 \rangle}$ for $n \gg 0$. Let \begin{align} H(R,n-1) = {k(n-1) \choose n-1}+{k(n-2) \choose n-2}+\cdots+{k(n-j) \choose n-j} \end{align} be the $(n-1)$-th Macaulay expansion of $H(R,n-1)$ for some large $n$. Then \begin{align} H(R,n-1)^{\langle n-1 \rangle} = {k(n-1)+1 \choose n}+{k(n-2)+1 \choose n-1}+\cdots+{k(n-j)+1 \choose n-j+1}. \end{align} Since $k(n-i) \ge n-i$ we have that $k(n-i)+1 = n-i+1+a_i$ for some non-negative integers $a_i$. Hence \begin{align} H(R,n) = H(R,n-1)^{\langle n-1 \rangle} &= {n+a_1 \choose n}+{n+a_2-1 \choose n-1}+\cdots+{n+a_j-(j-1) \choose n-j+1} \\ &= {n+a_1 \choose a_1}+{n+a_2-1 \choose a_2}+\cdots+{n+a_j-(j-1) \choose a_j}.\end{align}
At first sight we are done. However, the quantities $a_i$ depend on $H(R,n-1)$, i.e., they are functions of $n$. On the other hand, the statement of the exercise implies implies that these quantities are constants. Recall that $a_i:=k(n-i)-(n-i)$; it is not clear to me why these should be constants. So how can we argue to establish the statement of the exercise?
