A262045 Irregular triangle read by rows in which row n lists the elements of row n of A249223 and then the elements of the same row in reverse order.

1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Hartmut F. W. Hoft, Sep 09 2015

Keywords

• nonn
• tabf

Comments

The n-th row of the triangle has length 2*A003056(n).

This sequence extends A249223 in the same manner as A237593 extends A237591.

The entries in the n-th row of the triangle are the widths of the regions between the (n-1)-st and n-th Dyck paths for the symmetric representation of sigma(n) with each column representing the corresponding leg of the n-th path.

Examples

			n\k 1   2   3   4   5   6   7   8   9   10
1   1   1
2   1   1
3   1   0   0   1
4   1   1   1   1
5   1   0   0   1
6   1   1   2   2   1   1
7   1   0   0   0   0   1
8   1   1   1   1   1   1
9   1   0   1   1   0   1
10  1   1   1   0   0   1   1   1
11  1   0   0   0   0   0   0   1
12  1   1   2   2   2   2   1   1
13  1   0   0   0   0   0   0   1
14  1   1   1   0   0   1   1   1
15  1   0   1   1   2   2   1   1   0   1
16  1   1   1   1   1   1   1   1   1   1
17  1   0   0   0   0   0   0   0   0   1
18  1   1   2   1   1   1   1   2   1   1
19  1   0   0   0   0   0   0   0   0   1
20  1   1   1   1   2   2   1   1   1   1
...
The triangle shows that the region between a Dyck path for n and n-1 has width 1 if n is a power of 2. For n a prime the region is a horizontal rectangle of width (height) 1 and the vertical rectangle of width 1 which is its reflection. The Dyck paths and regions are shown below for n = 1..5 (see the A237593 example for n = 1..28):
   _ _ _
5 |_ _ _|
4 |_ _  |_ _
3 |_ _|_  | |
2 |_  | | | |
1 |_|_|_|_|_|
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..3200 [First 150 rows, based on G. C. Greubel's b-file for A249223]

Crossrefs

Cf. A000203, A003056, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A249223, A262048.

Programs

Mathematica

(* functions a237048[ ] and row[ ] are defined in A237048 *)
f[n_] :=Drop[FoldList[Plus, 0, Map[(-1)^(#+1)&, Range[row[n]]] a237048[n]], 1]
a262045[n_]:=Join[f[n], Reverse[f[n]]]
Flatten[Map[a262045, Range[16]]](* data *)

Formula

T(n, k) = T(n, 2*A003056(n) + 1 - k) = A249223(n, k), for 1 <= n and 1 <= k <= A003056(n).