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A265507 A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k). 0
1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1, 0, 5, 0, 0, 2, 0, 4, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 8, 0, 1, 0, 9, 0, 10, 0, 2, 0, 8, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 13, 0, 0, 1, 0, 13, 0, 36, 0, 0, 3, 0, 23, 0, 24, 0, 0, 3, 0, 12, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

A positive arch is defined as a top arch that starts at an odd-numbered vertex and ends at a higher even-numbered vertex.

For each value of n there is a square array with n^2 elements.

Rows are in order of decreasing number of components.

The sum of all the elements in each square array(n) = Catalan numbers C(n) A000108.

The sum of columns for array(n) = Semimeander components row(n) A046726.

The sum of the rows for array(n) = Narayana numbers T(n,k) A001263.

All semimeander solutions (k=1) for array n have positive arches = floor((n+2)/2).

LINKS

Table of n, a(n) for n=1..85.

EXAMPLE

For n=3:                                   /\          /\

               /\               /\        /  \        //\\

              /  \             /  \      /    \      //  \\

  /\ /\ /\   / /\ \  /\   /\  / /\ \    //\  /\\    // /\ \\

  \ \\// /   \ \ \/ / /   \ \ \/ / /    \\ \/ //    \\ \/ //

   \ \/ /     \ \  / /     \ \  / /      \\  //      \\  //

    \  /       \ \/ /       \ \/ /        \\//        \\//

     \/         \  /         \  /          \/          \/

                 \/           \/

  p=3,k=2     p=2,k=1      p=2,k=1      p=1,k=2     p=2,k=3.

.

n=3  p\k 3  2  1   n=9  p\k 9  8  7  6  5  4  3  2  1

      1: 0  1  0         1: 0  0  0  0  1  0  0  0  0

      2: 1  0  2         2: 0  0  0  4  0 32  0  0  0

      3: 0  1  0         3: 0  0  6  0 78  0 252 0  0

                         4: 0  4  0 72  0 446 0 654 0

                         5: 1  0 29  0 280 0 950 0 504

                         6: 0  4  0 72  0 446 0 654 0

                         7: 0  0  6  0 78  0 252 0  0

                         8: 0  0  0  4  0 32  0  0  0

                         9: 0  0  0  0  1  0  0  0  0

CROSSREFS

Cf. A000108, A001263, A046726.

Sequence in context: A123858 A193261 A283497 * A035145 A244315 A214303

Adjacent sequences:  A265504 A265505 A265506 * A265508 A265509 A265510

KEYWORD

nonn,tabf

AUTHOR

Roger Ford, Dec 09 2015

STATUS

approved

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Last modified September 18 02:19 EDT 2020. Contains 337164 sequences. (Running on oeis4.)