OFFSET
0,5
COMMENTS
M also satisfies: [M^(4k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(4^n)](i,j) = M(i+n,j+n) for all n>=0. Conjecture: sum of the n-th row equals the partitions of 4^n into powers of 4.
FORMULA
M(n, k) = the coefficient of x^(4^n - 4^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(4^j)) whenever 0<=k<n for all n>0 (conjecture).
EXAMPLE
The 4th power of matrix is the same matrix excluding the first row and column:
[1,__0,__0,_0,0]^4=[____1,____0,___0,__0,0]
[1,__1,__0,_0,0]___[____4,____1,___0,__0,0]
[1,__4,__1,_0,0]___[___28,___16,___1,__0,0]
[1,_28,_16,_1,0]___[__524,__496,__64,__1,0]
[1,524,496,64,1]___[29804,41136,8128,256,1]
MATHEMATICA
dim = 9;
a[_, 0] = 1; a[i_, i_] = 1; a[i_, j_] /; j > i = 0;
M = Table[a[i, j], {i, 0, dim-1}, {j, 0, dim-1}];
M4 = MatrixPower[M, 4];
sol = Table[M4[[i, j]] == M[[i+1, j+1]], {i, 1, dim-1}, {j, 1, dim-1}] // Flatten // Solve;
Table[a[i, j], {i, 0, dim-1}, {j, 0, i}] /. sol // Flatten (* Jean-François Alcover, Oct 20 2019 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 29 2002
EXTENSIONS
More terms from Jean-François Alcover, Oct 20 2019
STATUS
approved