Question Number 126543 by pticantor last updated on 21/Dec/20
$$\:\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\underset{{i}=\mathrm{1}} {\overset{{k}} {\prod}}{i}^{{k}} =??? \\ $$
Answered by MJS_new last updated on 21/Dec/20
$$=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left({k}!\right)^{{k}} \\ $$$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{this}\:\mathrm{can}\:\mathrm{be}\:\mathrm{further}\:\mathrm{simplified} \\ $$$$\mathrm{you}\:\mathrm{could}\:\mathrm{write}\:\mathrm{it}\:\mathrm{as} \\ $$$$\mathrm{1}!+\left(\mathrm{2}!\right)^{\mathrm{2}} +\left(\mathrm{3}!\right)^{\mathrm{3}} +…\left({n}!\right)^{{n}} \\ $$