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A193092 Augmentation of the triangular array P given by p(n,k)=k! for 0<=k<=n. See Comments.

%I A193092 #9 Mar 30 2012 18:57:38
%S A193092 1,1,1,1,2,3,1,3,7,13,1,4,12,32,69,1,5,18,58,173,421,1,6,25,92,321,
%T A193092 1058,2867,1,7,33,135,523,1977,7159,21477,1,8,42,188,790,3256,13344,
%U A193092 53008,175769,1,9,52,252,1134,4986,21996,97956,427401,1567273
%N A193092 Augmentation of the triangular array P given by p(n,k)=k! for 0<=k<=n.  See Comments.
%C A193092 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193092 Regarding W=A193092, we have w(n,n)=A088368.
%e A193092 First 7 rows:
%e A193092 1
%e A193092 1...1
%e A193092 1...2....3
%e A193092 1...3....7.....13
%e A193092 1...4....12....32....69
%e A193092 1...5....18....58....173...421
%e A193092 1...6....25....92....321...1058...2867
%e A193092 The matrix method described at A193091 shows that row 3 arises from row 2 as the matrix product
%e A193092 ............. (1...1...2...4)
%e A193092 (1...2...3) * (0...1...1...2) = (1...3...7...13)
%e A193092 ............. (0...0...1...1).
%e A193092 The equivalent polynomial substitution method:
%e A193092 x^2+2x+3 -> (x^3+x^2+2x+4)+2(x^2+x+2)+3(x+1)= x^3+3x^2+7x+13.
%t A193092 p[n_, k_] := k!
%t A193092 Table[p[n, k], {n, 0, 5}, {k, 0, n}]
%t A193092 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193092 TableForm[m[4]]
%t A193092 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193092 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193092 v[n_] := v[n - 1].m[n]
%t A193092 TableForm[Table[v[n], {n, 0, 6}]] (* A193092 *)
%t A193092 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193092 Cf. A088368, A193091.
%K A193092 nonn,tabl
%O A193092 0,5
%A A193092 _Clark Kimberling_, Jul 30 2011