∑i=12n−1(nfi)=∑i=1n(ni)2n−i=∑i=1n(ni)2n−i组合意义,3种颜色染色。=3n−2n
\begin{aligned}
\sum_{i = 1}^{2^n - 1} \binom{n}{f_i} &= \sum_{i = 1}^{n} \binom{n}{i} 2^{n - i}
\\
&= \sum_{i = 1}^{n} \binom{n}{i}2^{n - i}
\\
&组合意义,3种颜色染色。
\\
&= 3^{n} - 2^{n}
\end{aligned}
i=1∑2n−1(fin)=i=1∑n(in)2n−i=i=1∑n(in)2n−i组合意义,3种颜色染色。=3n−2n
∑i=1n(ni)2n−i=∑i=0n−1(ni)2i=−2n+∑i=0n(ni)2i=−2n+(2+1)n=−2n+3n=3n−2n
\begin{aligned}
\sum_{i = 1}^{n} \binom{n}{i}2^{n - i} &= \sum_{i = 0}^{n - 1} \binom{n}{i} 2^i
\\
&= -2^{n} + \sum_{i = 0}^{n} \binom{n}{i}2^i
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&= -2^{n} + (2+1)^{n}
\\
&= -2^{n} + 3^{n}
\\
&= 3^n - 2^n
\end{aligned}
i=1∑n(in)2n−i=i=0∑n−1(in)2i=−2n+i=0∑n(in)2i=−2n+(2+1)n=−2n+3n=3n−2n
∑i=12n−1gifi(nfi)=∑i=1n2n−i(2i−1)(ni)i=∑i=1n \begin{aligned} \sum_{i = 1}^{2^n - 1} g_i f_i \binom{n}{f_i} &= \sum_{i = 1}^{n} 2^{n - i} (2^{i} - 1) \binom{n}{i} i \\ &= \sum_{i = 1}^{n} \end{aligned} i=1∑2n−1gifi(fin)=i=1∑n2n−i(2i−1)(in)i=i=1∑n