From: Eugene Gath * <eugene.gath@ul.ie>
To: rusin <rusin@math.niu.edu>
Subject: Re: Polynomial expansion
Date: Tue, 02 Jul 96 17:37:00 GMT

>In article <gathe.56.01507031@ul.ie>, Eugene Gath <gathe@ul.ie> wrote:
>>Define the n th degree polynomial p_n (x) := (x+1)(x+2)...(x+n).
>>Now expand this to obtain p_n (x)= sum_{i=0}^n a_i x^i.
>>Clearly a_0=n!, a_{n} = 1, a_{n-1} =1/2 n(n+1) etc.
>>In general a_{n-i}= sum(1 =< j_1 < j_2<....< j_i <= n) j_1 j_2 ... j_i
>>My question: is there a closed form for a_i?
>
>You are asking for the elementary symmetric functions of  {1, 2, ..., n}.
>There are comparatively compact formulae for the symmetric functions
>        s_i = 1^i + 2^i + ... + n^i

I don't see that these are directly relevant here, except perhaps through 
some combinations?
The answer is that the a_i s are the Stirling numbers of the first kind, 
which was pointed out to me by Bill Dubuque. There's a discussion about them 
in Knuth's Concrete Mathematics.
Thanks for your help.

Eugene Gath