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Question Number 126543 by pticantor last updated on 21/Dec/20

 Σ_(k=1) ^n Π_(i=1) ^k i^k =???

$$\:\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\underset{{i}=\mathrm{1}} {\overset{{k}} {\prod}}{i}^{{k}} =??? \\ $$

Answered by MJS_new last updated on 21/Dec/20

=Σ_(k=1) ^n (k!)^k   I don′t think this can be further simplified  you could write it as  1!+(2!)^2 +(3!)^3 +...(n!)^n

$$=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left({k}!\right)^{{k}} \\ $$$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{this}\:\mathrm{can}\:\mathrm{be}\:\mathrm{further}\:\mathrm{simplified} \\ $$$$\mathrm{you}\:\mathrm{could}\:\mathrm{write}\:\mathrm{it}\:\mathrm{as} \\ $$$$\mathrm{1}!+\left(\mathrm{2}!\right)^{\mathrm{2}} +\left(\mathrm{3}!\right)^{\mathrm{3}} +...\left({n}!\right)^{{n}} \\ $$

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