OFFSET
0,2
COMMENTS
For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding W=A193093:
col 1: A000142, n!
col 2: A001593, n*n!
col 3: A130744, n*(n+2)*n!
diag (1,1,3,13,71,...): A003319, indecomposable permutations.
It appears that T(n,k) is the number of indecomposable permutations p of [n+2] for which p(k+2) = 1. For example, T(2,1) = 4 counts 2413, 3412, 4213, 4312. - David Callan, Aug 27 2014
EXAMPLE
First 5 rows:
1
2.....1
6.....4....3
24....18...16...13
120...96...90...84...71
MATHEMATICA
p[n_, k_] := If[k == 0, n + 1, 1]
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A130296 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193094 *)
Flatten[Table[v[n], {n, 0, 9}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 30 2011
STATUS
approved