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

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, 1058, 2867, 1, 7, 33, 135, 523, 1977, 7159, 21477, 1, 8, 42, 188, 790, 3256, 13344, 53008, 175769, 1, 9, 52, 252, 1134, 4986, 21996, 97956, 427401, 1567273

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=A193092, we have w(n,n)=A088368.

Examples

			First 7 rows:
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...1058...2867
The matrix method described at A193091 shows that row 3 arises from row 2 as the matrix product
............. (1...1...2...4)
(1...2...3) * (0...1...1...2) = (1...3...7...13)
............. (0...0...1...1).
The equivalent polynomial substitution method:
x^2+2x+3 -> (x^3+x^2+2x+4)+2(x^2+x+2)+3(x+1)= x^3+3x^2+7x+13.
		

Crossrefs

Cf. A088368, A193091.

Programs

Mathematica

p[n_, k_] := k!
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
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}]] (* A193092 *)
Flatten[Table[v[n], {n, 0, 8}]]