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# 伯努利数

[begin{aligned}\sum_{i=0}^{m}{binom{m+1} i}B_i=[m=0],end{aligned}
]

[sum_{i=0}^{m}frac{B_i}{i!}frac{1}{(m-i)!}=frac{[m=1]}{m!}+frac{B_m}{m!},
]

[B(x)e^x=x+B(x),
]

[B(x)=frac{x}{e^x-1}.
]

# 自然数幂和

[S_m(n)=sum_{i=0}^{n-1}i^m,
]

[S_m(n)=frac{1}{m+1}sum_{i=1}^{m+1}{binom {m+1}i}B_{m+1-i}n^i,
]

[begin{aligned}sum_{ige 0}frac{S_m(n)x^m}{m!}&=sum_{mge 0}frac{x^m}{m!}sum_{i=0}^{n-1}i^m\&=sum_{i=0}^{n-1}sum_{mge 0}frac{x^mi^m}{m!}\&=sum_{i=0}^{n-1}e^{xi}\&=frac{e^{nx}-1}{e^x-1},end{aligned}
]

[frac{e^{nx}-1}{e^x-1}=frac{x}{e^x-1}×frac{e^{nx}-1}{x},
]

[sum_{ige 0}frac{S_i(n)x^i}{i!}=sum_{ige 0}frac{B_ix^i}{i!}sum_{ige 0}frac{n^{i+1}x^{i}}{(i+1)!},
]

[begin{aligned}[frac{x^m}{m!}]sum_{ige 0}frac{S_i(n)x^i}{i!}&=m!sum_{i=0}^mfrac{B_i}{i!}frac{n^{m+1-i}}{(m+1-i)!}\&=frac{1}{m+1}sum_{i=0}^m{binom {m+1} i}B_in^{m+1-i}\&=frac{1}{m+1}sum_{i=1}^{m+1}{binom {m+1} i}B_{m+1-i}n^i,end{aligned}
]

[S_m(n)=frac{1}{m+1}sum_{i=1}^{m+1}{binom {m+1} i}B_{m+1-i}n^i.
]