A193561 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(list; table; graph; refs; listen; history; text; internal format)

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:

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