determinant of a curious class of (0,1) matrices

[Michael Somos]

Marc LeBrun wrote :

> After making a few quick experiments I believe it to be true because it 
> appears to be true of any {0,1} matrix for which m(i,j)=0 when i+j is 
> odd.  Perhaps this weaker condition would be easier to prove.

Actually, more is true. I came across this naturally in my work on
"Number Walls in Combinatorics" where I looked at the number wall of
a sequence with zeros and the sequence without zeros. There is a
simple relationship between the entries of the two number walls.
They can be written in PARI-GP notation as follows :

a(n)=if(n<1,0,eval(Str("a"n)))
b(n)=if(n<1|n%2,0,eval(Str("a"n/2)))

N(n,m)=matdet(matrix(n,n,i,j,a(i+j+m-n-1)))
M(n,m)=matdet(matrix(n,n,i,j,b(i+j+m-n-1)))

M(2*n-1,2*m-1)==0
M(2*n-1,2*m)==N(n,m)*N(n-1,m)
M(2*n,2*m-1)==N(n,m)*N(n,m-1)
M(2*n,2*m)==(-1)^n*N(n,m)^2

M(i,j)==if(i%2==1&j%2==1,0,if(i%2==0&j%2==0,(-1)^(i/2),1)*\
           N(floor(i/2),floor(j/2))*N(ceil(i/2),ceil(j/2)))

Shalom, Michael