A352460 Triangle read by rows: T(n,k), 2 <= k < n is the number of n-element k-ary unlabeled quasi-full rooted trees.

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 1, 1, 9, 6, 5, 3, 2, 1, 1, 13, 10, 6, 5, 3, 2, 1, 1, 23, 15, 10, 7, 5, 3, 2, 1, 1, 35, 24, 14, 10, 7, 5, 3, 2, 1, 1, 61, 39, 23, 14, 11, 7, 5, 3, 2, 1, 1, 98, 63, 34, 21, 14, 11, 7, 5, 3, 2, 1, 1

Offset

3,4

Comments

A t-ary rooted tree is called quasi-full if the root of every subtree with r elements (nodes or vertices) has exactly min(r-1,t) children.

Links

Table of n, a(n) for n=3..80.

Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, Recognition and Enumeration of the Quasi-Full Rooted Trees, Journal of Integer Sequences, Vol. 28 (2025), Article 25.6.3. See pp. 2, 17.

Example

Triangle begins:
    1;
    1,  1;
    2,  1,  1;
    2,  2,  1,  1;
    4,  3,  2,  1,  1;
    5,  4,  3,  2,  1,  1;
    9,  6,  5,  3,  2,  1, 1;
   13, 10,  6,  5,  3,  2, 1, 1;
   23, 15, 10,  7,  5,  3, 2, 1, 1;
   35, 24, 14, 10,  7,  5, 3, 2, 1, 1;
   61, 39, 23, 14, 11,  7, 5, 3, 2, 1, 1;
   98, 63, 34, 21, 14, 11, 7, 5, 3, 2, 1, 1;
In particular, the rooted trees counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:
  ---------
    o   o
     \ /
      o
  ----------------------
    o       |
    |       |
    o   o   |   o  o  o
     \ /    |    \ | /
      o     |      o
  ------------------------------------------------------
    o   o        o   o   |   o         |
     \ /         |   |   |   |         |
      o   o      o   o   |   o  o  o   |   o  o   o  o
       \ /        \ /    |    \ | /    |    \  \ /  /
        o          o     |      o      |        o

Crossrefs

Cf. A000081, A000598, A001190, A292556, A299038.

Sequence in context: A106476 A101566 A331447 * A342767 A176653 A174842

Adjacent sequences: A352457 A352458 A352459 * A352461 A352462 A352463

Keyword

nonn,tabl

Author

Salah Uddin Mohammad, Mar 17 2022

Status

approved