A recurrence relation

I noticed that

(x_1 - x_k)\displaystyle\sum_{i_1+\cdots+i_k=n} x_1^{i_1}\cdots x_k^{i_k} = \displaystyle\sum_{i_1+\cdots+i_{k-1}=n+1} x_1^{i_1}\cdots x_{k-1}^{i_{k-1}} - \displaystyle\sum_{i_2+\cdots+i_k=n+1} x_2^{i_2}\cdots x_k^{i_k}.

In the difference on the RHS, it is apparent that terms without x_1 or x_k will vanish. Thus, all the negative terms which are not cancelled out have a x_k and all such positive terms have a x_1. Combinatorially, all terms of degree n+1 with x_k can be generated by multiplying x_k on all terms of degree n. Analogous holds for the positive terms. The terms with only x_1 and x_k are cancelled out with the exception of the x_1^{n+1} - x_k^{n+1} that remains.

This recurrence appears in calculation of the determinant of the Vandermonde matrix.