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Question Number 52669 by maxmathsup by imad last updated on 11/Jan/19

let S_(n )  (p)=Σ_(k=0) ^n  k^p   prove that S_n (p)=(1/(p+1)){ (n+1)^(p+1)  −Σ_(k=0) ^(n−1)  C_(p+1) ^k  S_n (k)}

$${let}\:{S}_{{n}\:} \:\left({p}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{{p}} \\ $$$${prove}\:{that}\:{S}_{{n}} \left({p}\right)=\frac{\mathrm{1}}{{p}+\mathrm{1}}\left\{\:\left({n}+\mathrm{1}\right)^{{p}+\mathrm{1}} \:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{C}_{{p}+\mathrm{1}} ^{{k}} \:{S}_{{n}} \left({k}\right)\right\} \\ $$

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