[seqfan] Re: Prove formula? Simple formula, difficult enumeration

[Richard Mathar]

http://list.seqfan.eu/pipermail/seqfan/2012-March/016671.html

rh> Date: Thu, 29 Mar 2012 07:12:21 -0700 (PDT)
rh> From: Ron Hardin 
rh> To: seqfan at list.seqfan.eu
rh> Subject: [seqfan] Prove formula?  Simple formula, difficult enumeration
rh> ...
rh> T(n,k)=Number of (n+1)X(n+1) -k..k symmetric matrices with every 2X2 subblock 
rh> having sum zero
rh> 
rh> Empirical: T(n,k)=k^(n+1)+(k+1)^(n+1)

Consider that the array is completely given once one has put 
a pattern of elements on the diagonal; that gives (2k)^(n+1)
patterns: 
 Two adjacent elements on the diagonal plus the requirement
 of zero sum fixes the two adjacent sub-diagonals, which in turn
 cascades further to the edges.
Now because of the symmetry one can only place either patterns
of even or odd elements on the diagonal, not mixed sets; otherwise
the adjacent diagonals would demand to be filled by fractions.
The sum k^(n+1) and (k+1)^(n+1) seems to count these two sets separately.  
This is to be considered a hand-waving, lack-of-time first step
towards a proof.

Richard Mathar