2乗と等比数列の積の和
\[\begin{aligned}
&S_n=\sum_{k=1}^n2^kk^2\\
\iff&
\left\{
\begin{array}{l}
S_{n+1}=S_n+2^{n+1}(n+1)^2\\
S_1=2\\
\end{array}
\right.
\\
\iff&S_{n+1}-2^{n+2}\{(n+1)^2-2(n+1)+3\}=S_n-2^{n+1}(n^2-2n+3)=S_1-8=-6\\
\iff&S_n=2^{n+1}(n^2-2n+3)-6
\end{aligned}\]