A078536 - OEIS

%I #10 Oct 20 2019 11:18:34

%S 1,1,1,1,4,1,1,28,16,1,1,524,496,64,1,1,29804,41136,8128,256,1,1,

%T 5423660,10272816,2755264,130816,1024,1,1,3276048300,8220685104,

%U 2804672704,178301696,2096128,4096,1,1,6744720496300,21934062166320,9139625620672,729250931456,11442760704,33550336,16384,1

%N 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.

%C 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.

%F 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).

%e The 4th power of matrix is the same matrix excluding the first row and column:

%e [1,__0,__0,_0,0]^4=[____1,____0,___0,__0,0]

%e [1,__1,__0,_0,0]___[____4,____1,___0,__0,0]

%e [1,__4,__1,_0,0]___[___28,___16,___1,__0,0]

%e [1,_28,_16,_1,0]___[__524,__496,__64,__1,0]

%e [1,524,496,64,1]___[29804,41136,8128,256,1]

%t dim = 9;

%t a[_, 0] = 1; a[i_, i_] = 1; a[i_, j_] /; j > i = 0;

%t M = Table[a[i, j], {i, 0, dim-1}, {j, 0, dim-1}];

%t M4 = MatrixPower[M, 4];

%t sol = Table[M4[[i, j]] == M[[i+1, j+1]], {i, 1, dim-1}, {j, 1, dim-1}] // Flatten // Solve;

%t Table[a[i, j], {i, 0, dim-1}, {j, 0, i}] /. sol // Flatten (* _Jean-François Alcover_, Oct 20 2019 *)

%Y Cf. A078121, A078122, A078537.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 29 2002

%E More terms from _Jean-François Alcover_, Oct 20 2019