Thursday, 13 October 2016

combinatorics - Is there a closed form expression for the following sum?



Is there a closed form expression for the following sum? $$\sum_{0\le i_1


I can understand that there are (nk) such terms and the values that i1++ik can take vary from k(k1)2 to k(nk)+k(k+1)2. So it remains to find how often the term kj=1ij=l is found as an exponent in the above sum. Can anyone give some idea? Thanks.


Answer



This is a particular case of a more general formula that I posted in here Sum of power functions over a simplex . By denoting your sum as Sk(n) we have:
Sk(n)=kj=0(1)jr12(j+k1)(kj)+j(n+1)(r;r)j(r;r)kj
where (r;r)j:=j1l=0(1rl+1).


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...