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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.

%I A193094 #13 Aug 27 2014 05:18:27
%S A193094 1,2,1,6,4,3,24,18,16,13,120,96,90,84,71,720,600,576,558,532,461,5040,
%T A193094 4320,4200,4128,4050,3908,3447,40320,35280,34560,34200,33888,33462,
%U A193094 32540,29093,362880,322560,317520,315360,313800,312096,309330,302436
%N 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.
%C A193094 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193094 Regarding W=A193093:
%C A193094 col 1:  A000142, n!
%C A193094 col 2:  A001593, n*n!
%C A193094 col 3:  A130744, n*(n+2)*n!
%C A193094 diag (1,1,3,13,71,...):  A003319, indecomposable permutations.
%C A193094 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 A193094 First 5 rows:
%e A193094 1
%e A193094 2.....1
%e A193094 6.....4....3
%e A193094 24....18...16...13
%e A193094 120...96...90...84...71
%t A193094 p[n_, k_] := If[k == 0, n + 1, 1]
%t A193094 Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A130296 *)
%t A193094 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193094 TableForm[m[4]]
%t A193094 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193094 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193094 v[n_] := v[n - 1].m[n]
%t A193094 TableForm[Table[v[n], {n, 0, 6}]] (* A193094 *)
%t A193094 Flatten[Table[v[n], {n, 0, 9}]]
%Y A193094 Cf. A193091, A130296, A193093.
%K A193094 nonn,tabl
%O A193094 0,2
%A A193094 _Clark Kimberling_, Jul 30 2011