%I #13 Aug 27 2014 05:18:27

%S 1,2,1,6,4,3,24,18,16,13,120,96,90,84,71,720,600,576,558,532,461,5040,

%T 4320,4200,4128,4050,3908,3447,40320,35280,34560,34200,33888,33462,

%U 32540,29093,362880,322560,317520,315360,313800,312096,309330,302436

%N Augmentation of the triangular array P=A130296 whose n-th row is (n+1,1,1,1,1...,1) for 0<=k<=n. See Comments.

%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

%C Regarding W=A193093:

%C col 1: A000142, n!

%C col 2: A001593, n*n!

%C col 3: A130744, n*(n+2)*n!

%C diag (1,1,3,13,71,...): A003319, indecomposable permutations.

%C 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

%e First 5 rows:

%e 1

%e 2.....1

%e 6.....4....3

%e 24....18...16...13

%e 120...96...90...84...71

%t p[n_, k_] := If[k == 0, n + 1, 1]

%t Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A130296 *)

%t m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]

%t TableForm[m[4]]

%t w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];

%t v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};

%t v[n_] := v[n - 1].m[n]

%t TableForm[Table[v[n], {n, 0, 6}]] (* A193094 *)

%t Flatten[Table[v[n], {n, 0, 9}]]

%Y Cf. A193091, A130296, A193093.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Jul 30 2011