A078536 Infinite lower triangular matrix, M, that satisfies [M^4](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

1, 1, 1, 1, 4, 1, 1, 28, 16, 1, 1, 524, 496, 64, 1, 1, 29804, 41136, 8128, 256, 1, 1, 5423660, 10272816, 2755264, 130816, 1024, 1, 1, 3276048300, 8220685104, 2804672704, 178301696, 2096128, 4096, 1, 1, 6744720496300, 21934062166320, 9139625620672, 729250931456, 11442760704, 33550336, 16384, 1

Offset: 0

Author

Paul D. Hanna, Nov 29 2002

Keywords

• nonn
• tabl

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.

Examples

			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]
		

Crossrefs

Cf. A078121, A078122, A078537.

Programs

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 *)

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<=k0 (conjecture).

Extensions

More terms from Jean-François Alcover, Oct 20 2019