RootSum Sequences

[Richard Guy]

No doubt you're aware of Newton's
theorem?  Essentially,

xP'(x)/P(x) is the generating
function.         R.

On Thu, 25 May 2006, Ed Pegg Jr wrote:

> Start with a polynomial with integer coefficients, and 
> find it's roots.  The sum of these roots, and the
> sums of the powers of these roots, is an integer.
>
> So, some examples, starting with x^3 - 6 x^2 + 4 x - 2.
>
> Table[RootSum[#^3 - 6 #^2 + 4 # - 2 &,#^n &],{n,1,20}]
>
> 6, 28, 150, 800, 4256, 22636, 120392, 640320, 3405624, 
> 18113248, 96337632, 512384048,
> 2725180256, 14494220608, 77089370720, 410009702400, 
> 2180689172736, 11598274968256,
> 61686912523392, 328089753612800
>
> Table[RootSum[4+12*#1-6*#1^2-8*#1^3+#1^4+#1^5&, 
> #^n&],{n,1,20}]
>
> {-1, 17, -7, 89, -51, 521, -379, 3201, -2851, 20137, 
> -21275, 128369, -156339, 825401,
> -1131147, 5340993, -8075171, 34736777, -57020027, 
> 226895889}
>
> Table[RootSum[4+4*#1-6*#1^2-#1^3+#1^4&, #^n&],{n,1,20}]
>
> {1,13,7,65,51,361,379,2081,2707,12233,18635,72881,124931,438745,822267,2663489,
> 5342387,16279273,34390571,100042705}
>
> Are there any polynomials that give some OEIS 
> Sequences?  As an aside, those last two polynomials
> are factors of the characteristic polynomial of the 
> Balaban 11-cage.  I wondered if the two polynomials
> were related in some strange way... and I then noticed 
> they shared {1, 7, 51, 379} in their sequences.
>
> Ed Pegg Jr.
>