总结一下公式，定期更新

∑ K = 0 N K C N K = N 2 N − 1 \sum_{K=0}^NKC_N^K=N2^{N-1} ∑K=0N​KCNK​=N2N−1

∑ K = 0 N ( K + 1 ) C N K = ( N + 2 ) 2 N − 1 \sum_{K=0}^N(K+1)C_N^K=(N+2)2^{N-1} ∑K=0N​(K+1)CNK​=(N+2)2N−1

∑ i = 0 k C n i C m k − i = C n + m k \sum_{i=0}^kC_n^iC_m^{k-i}=C_{n+m}^k ∑i=0k​Cni​Cmk−i​=Cn+mk​

C n m = C n − 1 m − 1 + C n m − 1 C_n^m=C_{n-1}^{m-1}+C_n^{m-1} Cnm​=Cn−1m−1​+Cnm−1​