#1900 · Algebra · Sequences & series

Sum of homogeneous products

Prove that the sum of the homogeneous products of $r$ dimensions of the numbers $1, 2, 3, \dots n$ , and their powers is $\frac{( - 1 ) ^{n - 1}}{( n - 1 )!} {1^{n + r - 1} - \frac{n - 1}{1} 2^{n + r - 1} + \frac{( n - 1 ) ( n - 2 )}{1 \cdot 2} 3^{n + r - 1} - \dots to n terms}$ .

~25 minProof · checked by AIFinite differencesGenerating functions

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Miscellaneous Examples, No. 295

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