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

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 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

Examples

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

Crossrefs

Cf. A193091.

Programs

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}]]