A078121 Infinite lower triangular matrix, M, that satisfies [M^2](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, 2, 1, 1, 4, 4, 1, 1, 10, 16, 8, 1, 1, 36, 84, 64, 16, 1, 1, 202, 656, 680, 256, 32, 1, 1, 1828, 8148, 10816, 5456, 1024, 64, 1, 1, 27338, 167568, 274856, 174336, 43680, 4096, 128, 1, 1, 692004, 5866452, 11622976, 8909648, 2794496, 349504, 16384, 256, 1

Offset

0,5

Comments

M also satisfies: [M^(2k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(2^n)](i,j) = M(i+n,j+n) for all n>=0.

Links

Alois P. Heinz, Rows n = 0..80, flattened

Formula

M(1,j) = A002577(j) (partitions of 2^j into powers of 2), M(j+1,j) = 2^j, M(j+2,j) = 4^j, M(j+3,j) = A016131(j).

M(n,k) = the coefficient of x^(2^n - 2^(n-k)) in the power series expansion of 1/Product_{j=0..n-k} (1-x^(2^j)) whenever 0<=k<n for all n>0 (conjecture).

Example

The square of the matrix is the same matrix excluding the first row and column:
  [1, 0, 0, 0, 0]^2 = [ 1, 0, 0, 0, 0]
  [1, 1, 0, 0, 0]     [ 2, 1, 0, 0, 0]
  [1, 2, 1, 0, 0]     [ 4, 4, 1, 0, 0]
  [1, 4, 4, 1, 0]     [10,16, 8, 1, 0]
  [1,10,16, 8, 1]     [36,84,64,16, 1]

Maple

M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(M(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(M(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015

Mathematica

rows = 10; M[k_] := Table[ Which[j == 1, 1, i == j, 1, 1 < j < i, m[i, j], True, 0], {i, 1, k}, {j, 1, k}]; m2[i_, j_] := m[i+1, j+1]; M2[k_] := Table[ Which[j<i, m2[i, j], j == i, 1, True, 0], {i, 1, k}, {j, 1, k}]; sol[k_] := Thread[ Flatten[ M[k].M[k]] == Flatten[M2[k]]] // Solve; Table[M[rows][[i, j]], {i, 1, rows}, {j, 1, i}] /. sol[rows] // Flatten (* Jean-François Alcover, Feb 27 2015 *)
M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[M[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[M[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 27 2015, after Alois P. Heinz *)

Crossrefs

Cf. A078122, A002577, A016131.

Sequence in context: A373744 A326326 A307139 * A333157 A119732 A260625

Adjacent sequences: A078118 A078119 A078120 * A078122 A078123 A078124

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 18 2002

Status

approved