A265507 A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k).

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

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