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m > 0 {\displaystyle m>0} , n > 0 {\displaystyle n>0} F ( n , m ) = ∑ k = 0 m ( − 1 ) k ( n k ) ( n − k m − k ) = 0 {\displaystyle F(n,m)=\sum _{k=0}^{m}(-1)^{k}{\binom {n}{k}}{\binom {n-k}{m-k}}=0}
どうやって証明するのかな。
F ( n , m ) = ∑ k = 0 m ( − 1 ) k [ ( n − 1 k − 1 ) + ( n − 1 k ) ] ( n − k m − k ) {\displaystyle F(n,m)=\sum _{k=0}^{m}(-1)^{k}\left[{\binom {n-1}{k-1}}+{\binom {n-1}{k}}\right]{\binom {n-k}{m-k}}}
= ∑ k = 0 m ( − 1 ) k ( n − 1 k − 1 ) ( n − k m − k ) + ∑ k = 0 m ( − 1 ) k ( n − 1 k ) ( n − k m − k ) {\displaystyle =\sum _{k=0}^{m}(-1)^{k}{\binom {n-1}{k-1}}{\binom {n-k}{m-k}}+\sum _{k=0}^{m}(-1)^{k}{\binom {n-1}{k}}{\binom {n-k}{m-k}}}
= ∑ k = 0 m ( − 1 ) k ( n − 1 k − 1 ) ( n − k m − k ) + ∑ k = 1 m + 1 ( − 1 ) k − 1 ( n − 1 k − 1 ) ( n − k + 1 m − k + 1 ) {\displaystyle =\sum _{k=0}^{m}(-1)^{k}{\binom {n-1}{k-1}}{\binom {n-k}{m-k}}+\sum _{k=1}^{m+1}(-1)^{k-1}{\binom {n-1}{k-1}}{\binom {n-k+1}{m-k+1}}}
= ( − 1 ) m ( n − 1 m ) + ∑ k = 1 m ( − 1 ) k ( n − 1 k − 1 ) ( n − k m − k ) − ∑ k = 1 m ( − 1 ) k ( n − 1 k − 1 ) ( n − k + 1 m − k + 1 ) {\displaystyle =(-1)^{m}{\binom {n-1}{m}}+\sum _{k=1}^{m}(-1)^{k}{\binom {n-1}{k-1}}{\binom {n-k}{m-k}}-\sum _{k=1}^{m}(-1)^{k}{\binom {n-1}{k-1}}{\binom {n-k+1}{m-k+1}}}
= ( − 1 ) m ( n − 1 m ) + ∑ k = 1 m ( − 1 ) k − 1 ( n − 1 k − 1 ) ( n − k m − k + 1 ) {\displaystyle =(-1)^{m}{\binom {n-1}{m}}+\sum _{k=1}^{m}(-1)^{k-1}{\binom {n-1}{k-1}}{\binom {n-k}{m-k+1}}}
= ∑ k = 1 m + 1 ( − 1 ) k − 1 ( n − 1 k − 1 ) ( n − k m − k + 1 ) {\displaystyle =\sum _{k=1}^{m+1}(-1)^{k-1}{\binom {n-1}{k-1}}{\binom {n-k}{m-k+1}}}
= ∑ k = 0 m ( − 1 ) k ( n − 1 k ) ( n − k − 1 m − k ) = F ( n − 1 , m ) {\displaystyle =\sum _{k=0}^{m}(-1)^{k}{\binom {n-1}{k}}{\binom {n-k-1}{m-k}}=F(n-1,m)}
n > 0 {\displaystyle n>0} , m > 0 {\displaystyle m>0} それに F ( n , m ) = F ( n − 1 , m ) {\displaystyle F(n,m)=F(n-1,m)} または F ( 0 , m ) = ∑ k = 0 m ( − 1 ) k ( 0 k ) ( − k m − k ) = 0 {\displaystyle F(0,m)=\sum _{k=0}^{m}(-1)^{k}{\binom {0}{k}}{\binom {-k}{m-k}}=0} であるから F ( n , m ) = 0 {\displaystyle F(n,m)=0}
, 
![{\displaystyle F(n,m)=\sum _{k=0}^{m}(-1)^{k}\left[{\binom {n-1}{k-1}}+{\binom {n-1}{k}}\right]{\binom {n-k}{m-k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71ee22468a98751cdbc15ebbd1ff9f087cc1bec)
, それに
または
であるから
