P1595信封问题
\[\begin{aligned}
n!=\sum\limits_{i=0}^{n}\binom{n}{i}f(i)\\
f(i)=\sum\limits_{i=0}^n(-1)^{n-i}\binom{n}{i}i!
\end{aligned}
\]
P10596集合计数
\[\begin{aligned}
\binom{N}{n}(2^{2^{N-n}}-1)=\sum\limits_{i=n}^N\binom{i}{n}f(i)\\
f(n)=\sum\limits_{i=n}^{N}(-1)^{i-n}\binom{i}{n}\binom{N}{i}(2^{2^{N-i}}-1)
\end{aligned}
\]
P5505分特产
\[\begin{aligned}
\binom{N}{n}\prod\limits_{i=1}^M\binom{a_i+N-1-i}{N-i-1}=\sum\limits_{i=n}^{N}\binom{i}{n}f(i)\\
f(n)=\sum\limits_{i=n}^{N}(-1)^{i-n}\binom{i}{n}\binom{N}{i}\prod\limits_{j=1}^M\binom{a_j+N-1-j}{N-j-1}
\end{aligned}
\]
P4859已经没有什么好害怕的了
\[\begin{aligned}
g(N, n)\times(N-n)!=\sum\limits_{i=n}^N\binom{i}{n}f(i)\\
f(n)=\sum\limits_{i=n}^N(-1)^{i-n}\binom{i}{n}g(N, i)\times(N-i)!\\
g(n, m)=g(n-1, m)+(sml_n-(m-1))g(n-1,m-1)
\end{aligned}
\]