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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,19

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 areas 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.

LINKS

Table of n, a(n) for n=1..101.

FORMULA

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

EXAMPLE

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 area a(n) between a Dyck path for n and n-1 has for n a power of 2 width 1. For n a prime the area a(n) is a horizontal rectangle of width (height) 1 and the vertical rectangle of width 1 needed by symmetry. The Dyck paths and areas are shown below for n=1,...,5 (see the A237593 example for n = 1..28):

   _ _ _

5 |_ _ _|

4 |_ _  |_ _

3 |_ _|_  | |

2 |_  | | | |

1 |_|_|_|_|_|

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 *)

CROSSREFS

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

Sequence in context: A004562 A261817 A123550 * A263087 A204433 A004578

Adjacent sequences:  A262042 A262043 A262044 * A262046 A262047 A262048

KEYWORD

nonn,tabf

AUTHOR

Hartmut F. W. Hoft, Sep 09 2015

STATUS

approved

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Last modified February 25 03:33 EST 2018. Contains 299630 sequences. (Running on oeis4.)