英文维基 | 中文维基 | 日文维基 | 草榴社区
u , w ∈ [ 0 , 1 ] {\displaystyle u,w\in [0,1]}
B ( u ) = ∑ k = 0 n ( n k ) u n − k ∑ j = 0 n − k ( − 1 ) j ( n − k j ) P n − k − j {\displaystyle B(u)=\sum _{k=0}^{n}{\binom {n}{k}}u^{n-k}\sum _{j=0}^{n-k}(-1)^{j}{\binom {n-k}{j}}P_{n-k-j}}
P ( l e f t , a ) = ∑ k = 0 a ( a k ) w a − k ∑ j = 0 a − k ( − 1 ) j ( a − k j ) P a − k − j {\displaystyle P_{(left,a)}=\sum _{k=0}^{a}{\binom {a}{k}}w^{a-k}\sum _{j=0}^{a-k}(-1)^{j}{\binom {a-k}{j}}P_{a-k-j}}
B l e f t ( u ) = ∑ k = 0 n ( n k ) u n − k ∑ j = 0 n − k ( − 1 ) j ( n − k j ) P ( l e f t , n − k − j ) {\displaystyle B_{left}(u)=\sum _{k=0}^{n}{\binom {n}{k}}u^{n-k}\sum _{j=0}^{n-k}(-1)^{j}{\binom {n-k}{j}}P_{(left,n-k-j)}}
= ∑ k = 0 n ( n k ) u n − k ∑ j = 0 n − k ( n − k j ) ( − 1 ) j ∑ i = 0 n − k − j ( n − k − j i ) w n − k − j − i ∑ h = 0 n − k − j − i ( n − k − j − i h ) ( − 1 ) h P ( n − k − j − i − h ) {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}u^{n-k}\sum _{j=0}^{n-k}{\binom {n-k}{j}}(-1)^{j}\sum _{i=0}^{n-k-j}{\binom {n-k-j}{i}}w^{n-k-j-i}\sum _{h=0}^{n-k-j-i}{\binom {n-k-j-i}{h}}(-1)^{h}P_{(n-k-j-i-h)}} = u n [ [ w n [ P n − n P n − 1 + ( n 2 ) P n − 2 − . . . + ( − 1 ) n − 2 ( n n − 2 ) P 2 + ( − 1 ) n − 1 n P 1 + ( − 1 ) n P 0 ] {\displaystyle =u^{n}{\Bigg [}{\bigg [}w^{n}[P_{n}-nP_{n-1}+{\tbinom {n}{2}}P_{n-2}-...+(-1)^{n-2}{\tbinom {n}{n-2}}P_{2}+(-1)^{n-1}nP_{1}+(-1)^{n}P_{0}]} + n w n − 1 [ P n − 1 − ( n − 1 ) P n − 2 + ( n − 1 2 ) P n − 3 − . . . + ( − 1 ) n − 3 ( n − 1 n − 3 ) P 2 + ( − 1 ) n − 2 ( n − 1 ) P 1 + ( − 1 ) n − 1 P 0 ] {\displaystyle +nw^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]} + ( n 2 ) w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +{\binom {n}{2}}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + . . . {\displaystyle +...} + ( n n − 2 ) w 2 [ P 2 − 2 P 1 + P 0 ] + n w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n}{n-2}}w^{2}[P_{2}-2P_{1}+P_{0}]+nw[P_{1}-P_{0}]+P_{0}{\bigg ]}}
− n [ w n − 1 [ P n − 1 − ( n − 1 ) P n − 2 + ( n − 1 2 ) P n − 3 − . . . + ( − 1 ) n − 3 ( n − 1 n − 3 ) P 2 + ( − 1 ) n − 2 ( n − 1 ) P 1 + ( − 1 ) n − 1 P 0 ] {\displaystyle -n{\bigg [}w^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]} + ( n − 1 ) w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +(n-1)w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + ( n − 1 2 ) w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +{\binom {n-1}{2}}w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + . . . {\displaystyle +...} + ( n − 1 n − 3 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 1 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-1}{n-3}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-1)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
+ ( n 2 ) [ w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +{\dbinom {n}{2}}{\bigg [}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + ( n − 2 ) w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +(n-2)w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + ( n − 2 2 ) w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +{\binom {n-2}{2}}w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + . . . {\displaystyle +...} + ( n − 2 n − 3 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 2 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-2}{n-3}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-2)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
− . . . {\displaystyle -...}
+ ( − 1 ) n − 2 ( n n − 2 ) [ w 2 [ P 2 − 2 P 1 + P 0 ] + 2 w [ P 1 − P 0 ] + P 0 ] {\displaystyle +(-1)^{n-2}{\dbinom {n}{n-2}}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+2w[P_{1}-P_{0}]+P_{0}{\bigg ]}} + ( − 1 ) n − 1 n [ w [ P 1 − P 0 ] + P 0 ] + ( − 1 ) n P 0 ] {\displaystyle +(-1)^{n-1}n{\bigg [}w[P_{1}-P_{0}]+P_{0}{\bigg ]}+(-1)^{n}P_{0}{\Bigg ]}}
+ n u n − 1 [ [ w n − 1 [ P n − 1 − ( n − 1 ) P n − 2 + ( n − 1 2 ) P n − 3 − . . . + ( − 1 ) n − 3 ( n − 1 n − 3 ) P 2 + ( − 1 ) n − 2 ( n − 1 ) P 1 + ( − 1 ) n − 1 P 0 ] {\displaystyle +nu^{n-1}{\Bigg [}{\bigg [}w^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]} + ( n − 1 ) w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +(n-1)w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + ( n − 1 2 ) w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +{\binom {n-1}{2}}w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + . . . {\displaystyle +...} + ( n − 1 n − 3 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 1 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-1}{n-3}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-1)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
− ( n − 1 ) [ w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle -(n-1){\bigg [}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + ( n − 2 ) w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +(n-2)w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + ( n − 2 2 ) w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +{\binom {n-2}{2}}w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + . . . {\displaystyle +...} + ( n − 2 n − 3 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 2 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-2}{n-3}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-2)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
+ ( n − 1 2 ) [ w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +{\dbinom {n-1}{2}}{\bigg [}w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + ( n − 3 ) w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +(n-3)w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + ( n − 3 2 ) w n − 5 [ P n − 5 − ( n − 5 ) P n − 6 + ( n − 5 2 ) P n − 7 − . . . + ( − 1 ) n − 7 ( n − 5 n − 7 ) P 2 + ( − 1 ) n − 6 ( n − 5 ) P 1 + ( − 1 ) n − 5 P 0 ] {\displaystyle +{\binom {n-3}{2}}w^{n-5}[P_{n-5}-(n-5)P_{n-6}+{\tbinom {n-5}{2}}P_{n-7}-...+(-1)^{n-7}{\tbinom {n-5}{n-7}}P_{2}+(-1)^{n-6}(n-5)P_{1}+(-1)^{n-5}P_{0}]} + . . . {\displaystyle +...} + ( n − 3 n − 4 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 3 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-3}{n-4}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-3)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
+ ( − 1 ) n − 3 ( n − 1 n − 3 ) [ w 2 [ P 2 − 2 P 1 + P 0 ] + 2 w [ P 1 − P 0 ] + P 0 ] {\displaystyle +(-1)^{n-3}{\dbinom {n-1}{n-3}}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+2w[P_{1}-P_{0}]+P_{0}{\bigg ]}} + ( − 1 ) n − 2 ( n − 1 ) [ w [ P 1 − P 0 ] + P 0 ] + ( − 1 ) n − 1 P 0 ] {\displaystyle +(-1)^{n-2}(n-1){\bigg [}w[P_{1}-P_{0}]+P_{0}{\bigg ]}+(-1)^{n-1}P_{0}{\Bigg ]}}
+ ( n 2 ) u n − 2 [ [ w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +{\dbinom {n}{2}}u^{n-2}{\Bigg [}{\bigg [}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + ( n − 2 ) w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +(n-2)w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + ( n − 2 2 ) w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +{\binom {n-2}{2}}w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + . . . {\displaystyle +...} + ( n − 2 n − 4 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 2 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-2}{n-4}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-2)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
− ( n − 2 ) [ w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle -(n-2){\bigg [}w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + ( n − 3 ) w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +(n-3)w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + ( n − 3 2 ) w n − 5 [ P n − 5 − ( n − 5 ) P n − 6 + ( n − 5 2 ) P n − 7 − . . . + ( − 1 ) n − 7 ( n − 5 n − 7 ) P 2 + ( − 1 ) n − 6 ( n − 5 ) P 1 + ( − 1 ) n − 5 P 0 ] {\displaystyle +{\binom {n-3}{2}}w^{n-5}[P_{n-5}-(n-5)P_{n-6}+{\tbinom {n-5}{2}}P_{n-7}-...+(-1)^{n-7}{\tbinom {n-5}{n-7}}P_{2}+(-1)^{n-6}(n-5)P_{1}+(-1)^{n-5}P_{0}]} + . . . {\displaystyle +...} + ( n − 3 n − 5 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 3 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-3}{n-5}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-3)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
+ ( n − 2 2 ) [ w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +{\dbinom {n-2}{2}}{\bigg [}w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + ( n − 4 ) w n − 5 [ P n − 5 − ( n − 5 ) P n − 6 + ( n − 5 2 ) P n − 7 − . . . + ( − 1 ) n − 7 ( n − 5 n − 7 ) P 2 + ( − 1 ) n − 6 ( n − 5 ) P 1 + ( − 1 ) n − 5 P 0 ] {\displaystyle +(n-4)w^{n-5}[P_{n-5}-(n-5)P_{n-6}+{\tbinom {n-5}{2}}P_{n-7}-...+(-1)^{n-7}{\tbinom {n-5}{n-7}}P_{2}+(-1)^{n-6}(n-5)P_{1}+(-1)^{n-5}P_{0}]} + ( n − 4 2 ) w n − 6 [ P n − 6 − ( n − 6 ) P n − 7 + ( n − 6 2 ) P n − 8 − . . . + ( − 1 ) n − 8 ( n − 6 n − 8 ) P 2 + ( − 1 ) n − 7 ( n − 6 ) P 1 + ( − 1 ) n − 6 P 0 ] {\displaystyle +{\binom {n-4}{2}}w^{n-6}[P_{n-6}-(n-6)P_{n-7}+{\tbinom {n-6}{2}}P_{n-8}-...+(-1)^{n-8}{\tbinom {n-6}{n-8}}P_{2}+(-1)^{n-7}(n-6)P_{1}+(-1)^{n-6}P_{0}]} + . . . {\displaystyle +...} + ( n − 4 n − 6 ) w 2 [ P 2 − 2 P 1 + P 0 ] + ( n − 4 ) w [ P 1 − P 0 ] + P 0 ] {\displaystyle +{\binom {n-4}{n-6}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-4)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}
+ ( − 1 ) n − 3 ( n − 2 n − 4 ) [ w 2 [ P 2 − 2 P 1 + P 0 ] + 2 w [ P 1 − P 0 ] + P 0 ] {\displaystyle +(-1)^{n-3}{\dbinom {n-2}{n-4}}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+2w[P_{1}-P_{0}]+P_{0}{\bigg ]}} + ( − 1 ) n − 2 ( n − 2 ) [ w [ P 1 − P 0 ] + P 0 ] + ( − 1 ) n − 1 P 0 ] {\displaystyle +(-1)^{n-2}(n-2){\bigg [}w[P_{1}-P_{0}]+P_{0}{\bigg ]}+(-1)^{n-1}P_{0}{\Bigg ]}}
+ . . . {\displaystyle +...}
+ ( n n − 2 ) u 2 [ [ w 2 [ P 2 − 2 P 1 + P 0 ] + 2 w [ P 1 − P 0 ] + P 0 ] − 2 [ w [ P 1 − P 0 ] + P 0 ] + P 0 ] + n u [ [ w 1 [ P 1 − P 0 ] + P 0 ] − P 0 ] + P 0 {\displaystyle +{\dbinom {n}{n-2}}u^{2}{\Bigg [}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+2w[P_{1}-P_{0}]+P_{0}{\bigg ]}-2{\bigg [}w[P_{1}-P_{0}]+P_{0}{\bigg ]}+P_{0}{\Bigg ]}+nu{\Bigg [}{\bigg [}w^{1}[P_{1}-P_{0}]+P_{0}{\bigg ]}-P_{0}{\Bigg ]}+P_{0}}
= u n [ w n [ P n − n P n − 1 + ( n 2 ) P n − 2 − . . . + ( − 1 ) n − 2 ( n n − 2 ) P 2 + ( − 1 ) n − 1 n P 1 + ( − 1 ) n P 0 ] {\displaystyle =u^{n}{\bigg [}w^{n}[P_{n}-nP_{n-1}+{\tbinom {n}{2}}P_{n-2}-...+(-1)^{n-2}{\tbinom {n}{n-2}}P_{2}+(-1)^{n-1}nP_{1}+(-1)^{n}P_{0}]} + [ n − n ] w n − 1 [ P n − 1 − ( n − 1 ) P n − 2 + ( n − 1 2 ) P n − 3 − . . . + ( − 1 ) n − 3 ( n − 1 n − 3 ) P 2 + ( − 1 ) n − 2 ( n − 1 ) P 1 + ( − 1 ) n − 1 P 0 ] {\displaystyle +[n-n]\ w^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]} + [ ( n 2 ) − n ( n − 1 ) + ( n 2 ) ] w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +[{\tbinom {n}{2}}-n(n-1)+{\tbinom {n}{2}}]\ w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + [ ( n 3 ) − n ( n − 1 2 ) + ( n − 2 ) ( n 2 ) − ( n 3 ) ] w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +[{\tbinom {n}{3}}-n{\tbinom {n-1}{2}}+(n-2){\tbinom {n}{2}}-{\tbinom {n}{3}}]\ w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + . . . {\displaystyle +...} + [ ( n n − 2 ) − n ( n − 1 n − 3 ) + ( n 2 ) ( n − 2 n − 4 ) − . . . + ( − 1 ) n − 4 ( n n − 4 ) ( 4 2 ) + 3 ( − 1 ) n − 3 ( n n − 3 ) + ( − 1 ) n − 2 ( n n − 2 ) ] w 2 [ P 2 − 2 P 1 + P 0 ] {\displaystyle +[{\tbinom {n}{n-2}}-n{\tbinom {n-1}{n-3}}+{\tbinom {n}{2}}{\tbinom {n-2}{n-4}}-...+(-1)^{n-4}{\tbinom {n}{n-4}}{\tbinom {4}{2}}+3(-1)^{n-3}{\tbinom {n}{n-3}}+(-1)^{n-2}{\tbinom {n}{n-2}}]\ w^{2}[P_{2}-2P_{1}+P_{0}]} + [ n − n ( n − 1 ) + ( n 2 ) ( n − 2 ) − . . . + 3 ( − 1 ) n − 3 ( n n − 3 ) + 2 ( − 1 ) n − 2 ( n n − 2 ) + ( − 1 ) n − 1 n ] w [ P 1 − P 0 ] {\displaystyle +[n-n(n-1)+{\tbinom {n}{2}}(n-2)-...+3(-1)^{n-3}{\tbinom {n}{n-3}}+2(-1)^{n-2}{\tbinom {n}{n-2}}+(-1)^{n-1}n]\ w[P_{1}-P_{0}]} + [ 1 − n + ( n 2 ) − . . . + ( − 1 ) n − 2 ( n n − 2 ) + ( − 1 ) n − 1 n + ( − 1 ) n ] P 0 ] {\displaystyle +[1-n+{\tbinom {n}{2}}-...+(-1)^{n-2}{\tbinom {n}{n-2}}+(-1)^{n-1}n+(-1)^{n}]\ P_{0}{\bigg ]}}
+ n u n − 1 [ w n − 1 [ P n − 1 − ( n − 1 ) P n − 2 + ( n − 1 2 ) P n − 3 − . . . + ( − 1 ) n − 3 ( n − 1 n − 3 ) P 2 + ( − 1 ) n − 2 ( n − 1 ) P 1 + ( − 1 ) n − 1 P 0 ] {\displaystyle +nu^{n-1}{\bigg [}w^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]} + [ ( n − 1 ) − ( n − 1 ) ] w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +[(n-1)-(n-1)]\ w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + [ ( n − 1 2 ) − ( n − 1 ) ( n − 2 ) + ( n − 1 2 ) ] w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +[{\tbinom {n-1}{2}}-(n-1)(n-2)+{\tbinom {n-1}{2}}]\ w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + [ ( n − 1 3 ) − ( n − 1 ) ( n − 2 2 ) + ( n − 3 ) ( n − 1 2 ) − ( n − 1 3 ) ] w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +[{\tbinom {n-1}{3}}-(n-1){\tbinom {n-2}{2}}+(n-3){\tbinom {n-1}{2}}-{\tbinom {n-1}{3}}]\ w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + . . . {\displaystyle +...} + [ ( n − 1 n − 3 ) − ( n − 1 ) ( n − 2 n − 4 ) + ( n − 1 2 ) ( n − 3 n − 5 ) − . . . + ( − 1 ) n − 5 ( n − 1 n − 5 ) ( 4 2 ) + 3 ( − 1 ) n − 4 ( n − 1 n − 4 ) + ( − 1 ) n − 3 ( n − 1 n − 3 ) ] w 2 [ P 2 − 2 P 1 + P 0 ] {\displaystyle +[{\tbinom {n-1}{n-3}}-(n-1){\tbinom {n-2}{n-4}}+{\tbinom {n-1}{2}}{\tbinom {n-3}{n-5}}-...+(-1)^{n-5}{\tbinom {n-1}{n-5}}{\tbinom {4}{2}}+3(-1)^{n-4}{\tbinom {n-1}{n-4}}+(-1)^{n-3}{\tbinom {n-1}{n-3}}]\ w^{2}[P_{2}-2P_{1}+P_{0}]} + [ ( n − 1 ) − ( n − 1 ) ( n − 2 ) + ( n − 1 2 ) ( n − 3 ) − . . . + 3 ( − 1 ) n − 4 ( n − 1 n − 4 ) + 2 ( − 1 ) n − 3 ( n − 1 n − 3 ) + ( − 1 ) n − 2 ( n − 1 ) ] w [ P 1 − P 0 ] {\displaystyle +[(n-1)-(n-1)(n-2)+{\tbinom {n-1}{2}}(n-3)-...+3(-1)^{n-4}{\tbinom {n-1}{n-4}}+2(-1)^{n-3}{\tbinom {n-1}{n-3}}+(-1)^{n-2}(n-1)]\ w[P_{1}-P_{0}]} + [ 1 − ( n − 1 ) + ( n − 1 2 ) − . . . + ( − 1 ) n − 3 ( n − 1 n − 3 ) + ( − 1 ) n − 2 ( n − 1 ) + ( − 1 ) n − 1 ] P 0 ] {\displaystyle +[1-(n-1)+{\tbinom {n-1}{2}}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}+(-1)^{n-2}(n-1)+(-1)^{n-1}]\ P_{0}{\bigg ]}}
+ ( n 2 ) u n − 2 [ w n − 2 [ P n − 2 − ( n − 2 ) P n − 3 + ( n − 2 2 ) P n − 4 − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) P 2 + ( − 1 ) n − 3 ( n − 2 ) P 1 + ( − 1 ) n − 2 P 0 ] {\displaystyle +{\tbinom {n}{2}}u^{n-2}{\bigg [}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]} + [ ( n − 2 ) − ( n − 2 ) ] w n − 3 [ P n − 3 − ( n − 3 ) P n − 4 + ( n − 3 2 ) P n − 5 − . . . + ( − 1 ) n − 5 ( n − 3 n − 5 ) P 2 + ( − 1 ) n − 4 ( n − 3 ) P 1 + ( − 1 ) n − 3 P 0 ] {\displaystyle +[(n-2)-(n-2)]\ w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]} + [ ( n − 2 2 ) − ( n − 2 ) ( n − 3 ) + ( n − 2 2 ) ] w n − 4 [ P n − 4 − ( n − 4 ) P n − 5 + ( n − 4 2 ) P n − 6 − . . . + ( − 1 ) n − 6 ( n − 4 n − 6 ) P 2 + ( − 1 ) n − 5 ( n − 4 ) P 1 + ( − 1 ) n − 4 P 0 ] {\displaystyle +[{\tbinom {n-2}{2}}-(n-2)(n-3)+{\tbinom {n-2}{2}}]\ w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]} + [ ( n − 2 3 ) − ( n − 2 ) ( n − 3 2 ) + ( n − 4 ) ( n − 2 2 ) − ( n − 2 3 ) ] w n − 5 [ P n − 5 − ( n − 5 ) P n − 6 + ( n − 5 2 ) P n − 7 − . . . + ( − 1 ) n − 7 ( n − 5 n − 7 ) P 2 + ( − 1 ) n − 6 ( n − 5 ) P 1 + ( − 1 ) n − 5 P 0 ] {\displaystyle +[{\tbinom {n-2}{3}}-(n-2){\tbinom {n-3}{2}}+(n-4){\tbinom {n-2}{2}}-{\tbinom {n-2}{3}}]\ w^{n-5}[P_{n-5}-(n-5)P_{n-6}+{\tbinom {n-5}{2}}P_{n-7}-...+(-1)^{n-7}{\tbinom {n-5}{n-7}}P_{2}+(-1)^{n-6}(n-5)P_{1}+(-1)^{n-5}P_{0}]} + . . . {\displaystyle +...} + [ ( n − 2 n − 4 ) − ( n − 2 ) ( n − 3 n − 5 ) + ( n − 2 2 ) ( n − 4 n − 6 ) − . . . + ( − 1 ) n − 6 ( n − 2 n − 6 ) ( 4 2 ) + 3 ( − 1 ) n − 5 ( n − 2 n − 5 ) + ( − 1 ) n − 4 ( n − 2 n − 4 ) ] w 2 [ P 2 − 2 P 1 + P 0 ] {\displaystyle +[{\tbinom {n-2}{n-4}}-(n-2){\tbinom {n-3}{n-5}}+{\tbinom {n-2}{2}}{\tbinom {n-4}{n-6}}-...+(-1)^{n-6}{\tbinom {n-2}{n-6}}{\tbinom {4}{2}}+3(-1)^{n-5}{\tbinom {n-2}{n-5}}+(-1)^{n-4}{\tbinom {n-2}{n-4}}]\ w^{2}[P_{2}-2P_{1}+P_{0}]} + [ ( n − 2 ) − ( n − 2 ) ( n − 3 ) + ( n − 2 2 ) ( n − 4 ) − . . . + 3 ( − 1 ) n − 5 ( n − 2 n − 5 ) + 2 ( − 1 ) n − 4 ( n − 2 n − 4 ) + ( − 1 ) n − 3 ( n − 2 ) ] w [ P 1 − P 0 ] {\displaystyle +[(n-2)-(n-2)(n-3)+{\tbinom {n-2}{2}}(n-4)-...+3(-1)^{n-5}{\tbinom {n-2}{n-5}}+2(-1)^{n-4}{\tbinom {n-2}{n-4}}+(-1)^{n-3}(n-2)]\ w[P_{1}-P_{0}]} + [ 1 − ( n − 2 ) + ( n − 2 2 ) − . . . + ( − 1 ) n − 4 ( n − 2 n − 4 ) + ( − 1 ) n − 3 ( n − 2 ) + ( − 1 ) n − 2 ] P 0 ] {\displaystyle +[1-(n-2)+{\tbinom {n-2}{2}}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}+(-1)^{n-3}(n-2)+(-1)^{n-2}]\ P_{0}{\bigg ]}}
+ ( n n − 2 ) u 2 [ w 2 [ P 2 − 2 P 1 + P 0 ] + [ 2 − 2 ] w [ P 1 − P 0 ] + [ 1 − 2 + 1 ] P 0 ] {\displaystyle +{\tbinom {n}{n-2}}u^{2}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+[2-2]\ w[P_{1}-P_{0}]+[1-2+1]\ P_{0}{\bigg ]}} + n u [ w [ P 1 − P 0 ] + [ 1 − 1 ] P 0 ] + P 0 {\displaystyle +nu{\bigg [}w[P_{1}-P_{0}]+[1-1]\ P_{0}{\bigg ]}+P_{0}}
= ∑ k = 0 n ( n k ) u n − k ∑ j = 0 n − k w n − k − j ( ∑ i = 0 n − k − j ( − 1 ) i ( n − k − j i ) P n − k − j − i ) ( ∑ h = 0 j ( − 1 ) h ( n − k h ) ( n − k − h j − h ) ) {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}u^{n-k}\sum _{j=0}^{n-k}w^{n-k-j}\left(\sum _{i=0}^{n-k-j}(-1)^{i}{\binom {n-k-j}{i}}P_{n-k-j-i}\right)\left(\sum _{h=0}^{j}(-1)^{h}{\binom {n-k}{h}}{\binom {n-k-h}{j-h}}\right)}
= ∑ k = 0 n ( n k ) u n − k w n − k ∑ i = 0 n − k ( − 1 ) i ( n − k i ) P n − k − i {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}u^{n-k}w^{n-k}\sum _{i=0}^{n-k}(-1)^{i}{\binom {n-k}{i}}P_{n-k-i}} [1]
= ∑ k = 0 n ( n k ) ( u w ) n − k ∑ j = 0 n − k ( − 1 ) j ( n − k j ) P n − k − j {\displaystyle =\sum _{k=0}^{n}{\binom {n}{k}}(uw)^{n-k}\sum _{j=0}^{n-k}(-1)^{j}{\binom {n-k}{j}}P_{n-k-j}}
= B ( u w ) {\displaystyle =B(uw)}
利用者:Beneficii/bezier2
![{\displaystyle u,w\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2b00c6ef4e8dd0fd9100f1505b8ab013dcd20a)
![{\displaystyle =u^{n}{\Bigg [}{\bigg [}w^{n}[P_{n}-nP_{n-1}+{\tbinom {n}{2}}P_{n-2}-...+(-1)^{n-2}{\tbinom {n}{n-2}}P_{2}+(-1)^{n-1}nP_{1}+(-1)^{n}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e46f5a205cbc9c9a7561b5457c6e0a30860fc686)
![{\displaystyle +nw^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc395d1513adbff3ece88050c5fb9793b1ce7f86)
![{\displaystyle +{\binom {n}{2}}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2137cbf8fbdab5d9220e6371fe008abcf580447b)
![{\displaystyle +{\binom {n}{n-2}}w^{2}[P_{2}-2P_{1}+P_{0}]+nw[P_{1}-P_{0}]+P_{0}{\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/591832839b9e78747e54507952da57d9f383c138)
![{\displaystyle -n{\bigg [}w^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2725fb006e7267c981a38bd16e699d4435a674)
![{\displaystyle +(n-1)w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e95f89f3ffbdd19855be8e2402ac804875d173e)
![{\displaystyle +{\binom {n-1}{2}}w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/172fb37992c3077340cd89b31a2060ef5764dc17)
![{\displaystyle +{\binom {n-1}{n-3}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-1)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3215a048bd6bc91aada34dd83c33e9cceff7392)
![{\displaystyle +{\dbinom {n}{2}}{\bigg [}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1300506cc8f595dbe98731ab275ccf53f10508)
![{\displaystyle +(n-2)w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99648c8b233249598db48e35033d541c77b675e)
![{\displaystyle +{\binom {n-2}{2}}w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/904799d4f18cf0d63a9acd53b576a831a251bffc)
![{\displaystyle +{\binom {n-2}{n-3}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-2)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79056c941df62524472b3a94a1107a7c6273a895)
![{\displaystyle +(-1)^{n-2}{\dbinom {n}{n-2}}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+2w[P_{1}-P_{0}]+P_{0}{\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d40d37b0e42b9c5a6ef337ea64b21f52bbfc6654)
![{\displaystyle +(-1)^{n-1}n{\bigg [}w[P_{1}-P_{0}]+P_{0}{\bigg ]}+(-1)^{n}P_{0}{\Bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/402fc6c7a495db5700010647a67a93181684013b)
![{\displaystyle +nu^{n-1}{\Bigg [}{\bigg [}w^{n-1}[P_{n-1}-(n-1)P_{n-2}+{\tbinom {n-1}{2}}P_{n-3}-...+(-1)^{n-3}{\tbinom {n-1}{n-3}}P_{2}+(-1)^{n-2}(n-1)P_{1}+(-1)^{n-1}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4b44da5da2aeeaf1a822944243c43aa185b5e7)
![{\displaystyle -(n-1){\bigg [}w^{n-2}[P_{n-2}-(n-2)P_{n-3}+{\tbinom {n-2}{2}}P_{n-4}-...+(-1)^{n-4}{\tbinom {n-2}{n-4}}P_{2}+(-1)^{n-3}(n-2)P_{1}+(-1)^{n-2}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac50d9ddbd0381d402c3705f5555cbabbf7848af)
![{\displaystyle +{\dbinom {n-1}{2}}{\bigg [}w^{n-3}[P_{n-3}-(n-3)P_{n-4}+{\tbinom {n-3}{2}}P_{n-5}-...+(-1)^{n-5}{\tbinom {n-3}{n-5}}P_{2}+(-1)^{n-4}(n-3)P_{1}+(-1)^{n-3}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adcf3c679c4cac2810b3f4ffd463c154760362fe)
![{\displaystyle +(n-3)w^{n-4}[P_{n-4}-(n-4)P_{n-5}+{\tbinom {n-4}{2}}P_{n-6}-...+(-1)^{n-6}{\tbinom {n-4}{n-6}}P_{2}+(-1)^{n-5}(n-4)P_{1}+(-1)^{n-4}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0714c84391c7df1998e2152353ba2cf1d0571e30)
![{\displaystyle +{\binom {n-3}{2}}w^{n-5}[P_{n-5}-(n-5)P_{n-6}+{\tbinom {n-5}{2}}P_{n-7}-...+(-1)^{n-7}{\tbinom {n-5}{n-7}}P_{2}+(-1)^{n-6}(n-5)P_{1}+(-1)^{n-5}P_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba65e7a5b46675abd644ae7c97421af1ab4d4029)
![{\displaystyle +{\binom {n-3}{n-4}}w^{2}[P_{2}-2P_{1}+P_{0}]+(n-3)w[P_{1}-P_{0}]+P_{0}{\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85e9e6ec391615ae6039bf438630547cc63a5199)
![{\displaystyle +(-1)^{n-3}{\dbinom {n-1}{n-3}}{\bigg [}w^{2}[P_{2}-2P_{1}+P_{0}]+2w[P_{1}-P_{0}]+P_{0}{\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf74b4ab1c5aba9bcf29b78282c96c6459939c48)
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