[seqfan] Guess the Recurrence

Ron Hardin rhhardin at att.net
Sat Dec 4 04:19:37 CET 2010

Computing T(n,k; z), the number of nXk 0..z arrays with every 2X2 subblock 
summing to 2z,
it comes out that there's a single recurrence for every n,k row or column for a 
given z

z=0 Empirical: a(n)=1*a(n-1)
z=1 Empirical: a(n)=3*a(n-1)-2*a(n-2)
z=2 Empirical: a(n)=6*a(n-1)-11*a(n-2)+6*a(n-3)
z=3 Empirical: a(n)=10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4)
z=4 Empirical: a(n)=15*a(n-1)-85*a(n-2)+225*a(n-3)-274*a(n-4)+120*a(n-5)
z=5 Empirical: 

(so the initial (z+1)X(z+1) of the T(n,k) array determines the whole thing)

The question is what's the system of the coefficients of the recurrences vs z?

The first coef is (z+1)(z+2)/2 and the last is (z+1)!

Can we guess the middle ones?

 rhhardin at mindspring.com
rhhardin at att.net (either)

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