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n-1-2n-2-3n-3-rn-r-n-n-1-n-1-n-2-n-3-n-r-1-n-1-2-n-2-3-n-3-4-n-4-r-n-r-1-0-n-1-n-2-n-3-n-Prove-the-above-identity-




Question Number 139560 by Dwaipayan Shikari last updated on 28/Apr/21
Σ_(n_1 +2n_2 +3n_3 +..+rn_r =n) ^n (1/(n_1 !n_2 !n_3 !..n_r !1^n_1  2^n_2  3^n_3  4^n_4  ...r^n_r  ))=1  0≥n_1 ,n_2 ,n_3 ,..≥n  Prove the above identity
$$\underset{{n}_{\mathrm{1}} +\mathrm{2}{n}_{\mathrm{2}} +\mathrm{3}{n}_{\mathrm{3}} +..+{rn}_{{r}} ={n}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}_{\mathrm{1}} !{n}_{\mathrm{2}} !{n}_{\mathrm{3}} !..{n}_{{r}} !\mathrm{1}^{{n}_{\mathrm{1}} } \mathrm{2}^{{n}_{\mathrm{2}} } \mathrm{3}^{{n}_{\mathrm{3}} } \mathrm{4}^{{n}_{\mathrm{4}} } …{r}^{{n}_{{r}} } }=\mathrm{1} \\ $$$$\mathrm{0}\geqslant{n}_{\mathrm{1}} ,{n}_{\mathrm{2}} ,{n}_{\mathrm{3}} ,..\geqslant{n} \\ $$$$\mathrm{P}{rove}\:\mathrm{t}{he}\:{above}\:{identity} \\ $$

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