[seqfan] Re: A rectangular array.

Max Alekseyev maxale at gmail.com
Thu Oct 25 08:53:22 CEST 2012


On Thu, Oct 25, 2012 at 1:18 AM, Ed Jeffery <lejeffery2 at gmail.com> wrote:
> T =
> [
> 1, 0, 1, 0, 1, ...
> 0, 2, 0, 0, 0, ...
> 0, 0, 3, 0, 0, ...
> 0, 0, 0, 4, 0, ...
> 0, 0, 0, 0, 5, ...
> ...
> ].
>
> The transpose of T could be submitted to OEIS as a triangle read by rows,
> viz., T = {{1}, {0,2}, {1,0,3}, ...}, because of its association with the
> following.
>
> Index the rows of array T by n = 1, 2, ..., and the columns by k = 1, 2,
> ..., and consider the m X m matrix A_m with entries [A_m]_{i,j} = T(i,j),
> i,j = 1, 2, ..., m.
>
> * It seems that the coefficients of the characteristic polynomial of A_m
> are given by the m-th row of A008276 [1] (Stirling numbers of the first
> kind). *
>
> (In fact, for * there is a comment, stated without proof, in A008276 to
> that effect; the proof seems difficult to me, and I would like to know it.)

This directly follows from the definition of the characteristic
polynomial as the determinant det(A_m - x*I_m) where I_m is the
identity matrix of size mxm.
Since (A_m - x*I_m) is an upper triangular matrix, its determinant is simply
(1-x)*(2-x)*(3-x)*...*(m-x) = (-1)^m * (x-1)*(x-2)*(x-3)*...*(x-m)
whose coefficients are Stirling numbers of the first kind.

Regards,
Max



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