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A193092 Augmentation of the triangular array P given by p(n,k)=k! for 0<=k<=n.  See Comments. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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: A102473 A011117 A069269 * A263484 A293985 A100324

Adjacent sequences:  A193089 A193090 A193091 * A193093 A193094 A193095

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jul 30 2011

STATUS

approved

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Last modified September 24 13:43 EDT 2021. Contains 347643 sequences. (Running on oeis4.)