Example 3.5 Find $\sum_{k=1}^{n} k^{3}$.


\[
\begin{aligned} 
\sum_{k=1}^{n} k^{3} & =\sum_{j=1}^{3}\binom{n+1}{j+1} \Delta^{j} O^{3} \\ 
& =\binom{n+1}{2} \cdot \Delta O^{3}+\binom{n+1}{3} \Delta^{2} O^{3}+\binom{n+1}{4} \Delta^{3} O^{3} \\ 
& =\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4} \\ 
& =\frac{n^{2}(n+1)^{2}}{4} .
\end{aligned}
\]