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

0,5

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.

Links

Table of n, a(n) for n=0..54.

Example

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.

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}]]

Crossrefs

Cf. A088368, A193091.

Sequence in context: A011117 A368401 A069269 * A263484 A293985 A100324

Adjacent sequences: A193089 A193090 A193091 * A193093 A193094 A193095

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 30 2011

Status

approved