or writing
(x - x1) (x - x2) ... (x - xn) = xn - a1xn-1 + a2xn-2 - ...,
this is
| 1 |
| 1 - a2 - 2a3 - 3a4 - ... - (n - 1) an |
Again, consider the general problem of “derangements.” We have to find the number of permutations such that exactly m of the letters are in places they originally occupied. We have the particular redundant product
(ax1 + x2 + ... + xn)ξ1 (x1 + ax2 + ... + xn)ξ2 ... (x1 + x2 + ... + axn)ξn
in which the sought number is the coefficient of am xξ11 xξ22 ... xξnn. The true generating function is derived from the determinant
| a | 1 | 1 | 1 | . | . | . |
| 1 | a | 1 | 1 | . | . | . |
| 1 | 1 | a | 1 | . | . | . |
| 1 | 1 | 1 | a | . | . | . |
| . | . | . | . | |||
| . | . | . | . |
and has the form
| 1 | . |
| 1 - aΣ x1 + (a - 1) (a + 1)Σ x1x2 - ... +(-)n (a - 1)n-1 (a + n - 1)x1x2 ... xn |