A193561 Augmentation of the triangle A004736. See Comments.

1, 2, 1, 6, 6, 3, 24, 36, 30, 15, 120, 240, 270, 210, 105, 720, 1800, 2520, 2520, 1890, 945, 5040, 15120, 25200, 30240, 28350, 20790, 10395, 40320, 141120, 272160, 378000, 415800, 374220, 270270, 135135, 362880, 1451520, 3175200, 4989600

Offset

0,2

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193561, if the triangle is written as (w(n,k)), then

w(n,n)=A001147(n), "double factorial numbers";

w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1))

col 1: A000142, n!

col 2: A001286, Lah numbers, (n-1)*n!/2

Links

Table of n, a(n) for n=0..39.

Example

First 5 rows of A193560:
1
2.....1
6.....6....3
24....36...30...15
120...240..270..210..105

Mathematica

p[n_, k_] := n + 1 - k
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A004736 *)
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}]]  (* A193561 *)
Flatten[Table[v[n], {n, 0, 8}]]

Crossrefs

Cf. A193091.

Sequence in context: A075181 A052121 A193895 * A328349 A117965 A111646

Adjacent sequences: A193558 A193559 A193560 * A193562 A193563 A193564

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 30 2011

Status

approved