F(x)=∑i=1⌊m2⌋(xi)(n−xm−i)i+∑i=⌊m2⌋+1m(xi)(n−xm−i)(m−i)=∑i=1⌊m2⌋xi(x−1i−1)(n−xm−i)i+∑i=⌊m2⌋+1m(xi)(n−x−1m−i−1)n−xm−i(m−i)=∑i=1⌊m2⌋x(x−1i−1)(n−xm−i)+∑i=⌊m2⌋+1m(xi)(n−x−1m−i−1)(n−x)=x∑i=1⌊m2⌋(x−1i−1)(n−xm−i)+(n−x)∑i=⌊m2⌋+1m(xi)(n−x−1m−i−1)=x∑i=1⌊m2⌋(x−1i−1)(n−xm−i)+(n−x)∑i=⌊m2⌋+1m(xi)(n−x−1m−i−1) \begin{aligned} \mathcal{F}(x) &= \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \binom {x}{i} \binom{n - x}{m - i} i + \sum_{i = \lfloor \frac{m}{2} \rfloor + 1}^{m} \binom {x}{i} \binom{n - x}{m - i} (m - i) \\ &= \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \frac{x}{i} \binom {x - 1}{i - 1} \binom{n - x}{m - i} i + \sum_{i = \lfloor \frac{m}{2} \rfloor + 1}^{m} \binom {x}{i} \binom{n - x - 1}{m - i - 1} \frac{n - x}{m - i} (m - i) \\ &= \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} x \binom {x - 1}{i - 1} \binom{n - x}{m - i} + \sum_{i = \lfloor \frac{m}{2} \rfloor + 1}^{m} \binom {x}{i} \binom{n - x - 1}{m - i - 1} (n - x) \\ &= x \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \binom {x - 1}{i - 1} \binom{n - x}{m - i} + (n - x) \sum_{i = \lfloor \frac{m}{2} \rfloor + 1}^{m} \binom {x}{i} \binom{n - x - 1}{m - i - 1} \\ &= x \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \binom {x - 1}{i - 1} \binom{n - x}{m - i} + (n - x) \sum_{i = \lfloor \frac{m}{2} \rfloor + 1}^{m} \binom {x}{i} \binom{n - x - 1}{m - i - 1} \end{aligned} F(x)=i=1∑⌊2m⌋(ix)(m−in−x)i+i=⌊2m⌋+1∑m(ix)(m−in−x)(m−i)=i=1∑⌊2m⌋ix(i−1x−1)(m−in−x)i+i=⌊2m⌋+1∑m(ix)(m−i−1n−x−1)m−in−x(m−i)=i=1∑⌊2m⌋x(i−1x−1)(m−in−x)+i=⌊2m⌋+1∑m(ix)(m−i−1n−x−1)(n−x)=xi=1∑⌊2m⌋(i−1x−1)(m−in−x)+(n−x)i=⌊2m⌋+1∑m(ix)(m−i−1n−x−1)=xi=1∑⌊2m⌋(i−1x−1)(m−in−x)+(n−x)i=⌊2m⌋+1∑m(ix)(m−i−1n−x−1)
令 G(x)=∑i=1⌊m2⌋(x−1i−1)(n−xm−i)\mathcal{G} (x) = \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \binom {x - 1}{i - 1} \binom{n - x}{m - i}G(x)=∑i=1⌊2m⌋(i−1x−1)(m−in−x)
G(x+1)=∑i=1⌊m2⌋(xi−1)(n−x−1m−i)=∑i=1⌊m2⌋xx−i+1(x−1i−1)n−x−m+in−x(n−xm−i)=G(x)−(x−1⌊m2⌋−1)(n−x−1m−⌊m2⌋−1) \begin{aligned} \mathcal{G} (x + 1) &= \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \binom {x}{i - 1} \binom{n - x - 1}{m - i} \\ &= \sum_{i = 1}^{\lfloor \frac{m}{2} \rfloor} \frac{x}{x - i + 1} \binom {x - 1}{i - 1} \frac{n - x - m + i}{n - x} \binom{n - x}{m - i} \\ &= \mathcal{G} (x) - \binom{x - 1}{\lfloor \frac{m}{2} \rfloor - 1}\binom{n - x - 1}{m - \lfloor \frac{m}{2} \rfloor - 1} \end{aligned} G(x+1)=i=1∑⌊2m⌋(i−1x)(m−in−x−1)=i=1∑⌊2m⌋x−i+1x(i−1x−1)n−xn−x−m+i(m−in−x)=G(x)−(⌊2m⌋−1x−1)(m−⌊2m⌋−1n−x−1)
同理 H(x)=∑i=⌊m2⌋+1m(xi)(n−x−1m−i−1)\mathcal{H} (x) = \sum_{i = \lfloor \frac{m}{2} \rfloor + 1}^{m} \binom {x}{i} \binom{n - x - 1}{m - i - 1}H(x)=∑i=⌊2m⌋+1m(ix)(m−i−1n−x−1)
H(x+1)=H(x)+(n−x−2⌊m2⌋−1)(xm−⌊m2⌋−1) \mathcal{H} (x + 1) = \mathcal{H} (x) + \binom{n - x - 2}{\lfloor \frac{m}{2} \rfloor - 1} \binom{x}{m - \lfloor \frac{m}{2} \rfloor - 1} H(x+1)=H(x)+(⌊2m⌋−1n−x−2)(m−⌊2m⌋−1x)
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