A193094 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.

1, 2, 1, 6, 4, 3, 24, 18, 16, 13, 120, 96, 90, 84, 71, 720, 600, 576, 558, 532, 461, 5040, 4320, 4200, 4128, 4050, 3908, 3447, 40320, 35280, 34560, 34200, 33888, 33462, 32540, 29093, 362880, 322560, 317520, 315360, 313800, 312096, 309330, 302436

Offset: 0

Author

Clark Kimberling, Jul 30 2011

Keywords

• nonn
• tabl

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

Examples

			First 5 rows:
1
2.....1
6.....4....3
24....18...16...13
120...96...90...84...71
		

Crossrefs

Cf. A193091, A130296, A193093.

Programs

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}]]