OFFSET
0,5
COMMENTS
M also satisfies: [M^(3k)](i,j) = [M^k](i+1,j+1) for all i,j,k >=0; thus [M^(3^n)](i,j) = M(i+n,j+n) for all n >= 0.
Conjecture: the sum of the n-th row equals the number of partitions of 3^n into powers of 3 (A078125).
LINKS
Alois P. Heinz, Rows n = 0..60, flattened
FORMULA
EXAMPLE
The cube of the matrix is the same matrix excluding the first row and column:
[1, 0, 0, 0]^3 = [ 1, 0, 0, 0]
[1, 1, 0, 0] [ 3, 1, 0, 0]
[1, 3, 1, 0] [12, 9, 1, 0]
[1,12, 9, 1] [93,117,27, 1]
MAPLE
S:= proc(i, j) option remember;
add(M(i, k)*M(k, j), k=0..i)
end:
M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
add(S(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
m[i_, j_] := m[i, j]=If[j==0||i==j, 1, m3[i-1, j-1]]; m2[i_, j_] := m2[i, j]=Sum[m[i, k]m[k, j], {k, j, i}]; m3[i_, j_] := m3[i, j]=Sum[m[i, k]m2[k, j], {k, j, i}]; Flatten[Table[m[i, j], {i, 0, 8}, {j, 0, i}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 18 2002
STATUS
approved