(4) Let $(1+x)^{16}=\sum_{i=0}^{16} a_{i} x^{i}$, then $\sum_{i=1}^{8} i a_{i}=$ $\qquad$
(4) 262144 Hint: Given: $a_{i}=\mathrm{C}_{16}^{i}(i=0,1, \cdots, 16)$, and $i a_{i}=$ $i \mathrm{C}_{16}^{i}=16 \mathrm{C}_{15}^{i-1}$, so
$$
\sum_{i=1}^{8} i a_{i}=\sum_{i=1}^{8} 16 \mathrm{C}_{15}^{i-1}=16 \sum_{k=0}^{7} \mathrm{C}_{15}^{k}=16 \times 2^{14}=2^{18}=262144 .
$$