A364757 The pyramidal array T(r,g,b) = (r+g+b)/((g+b)*(r+b))*C(r+g,b-1)*C(g+b,r)*C(r+b,g), where 1 <= b <= ceiling((r+g+b)/2) and 0 <= r,g <= floor((r+g+b)/2). Read first over the layers corresponding to fixed sum r+g+b, then over the diagonals corresponding to fixed b.

1, 1, 1, 3, 1, 1, 2, 2, 1, 8, 1, 5, 15, 15, 1, 5, 1, 3, 3, 8, 54, 8, 1, 27, 27, 1, 7, 70, 70, 42, 168, 42, 1, 14, 14, 1, 4, 4, 30, 192, 30, 20, 400, 400, 20, 1, 64, 200, 64, 1, 9, 210, 210, 405, 1500, 405, 90, 900, 900, 90, 1, 30, 81, 30, 1, 5, 5, 80, 500, 80, 147, 2625, 2625, 147, 40, 1750, 5000, 1750, 40, 1, 125, 875, 875, 125, 1

Offset

1,4

Comments

T(r,g,b) is the number of injectively 3-colored trees with r red vertices, g green vertices, and b blue vertices, including a root vertex which is colored blue.

Summing T(r,g,b) over all r,g,b such that r+g+b=n yields the n-th Catalan number, A000108(n).

Column (or row) sums within each fixed r+g+b=n layer yield the number of ordered trees on n edges containing a fixed number of nodes adjacent to a leaf, A108759(n).

Main antidiagonal (corresponding to maximal value b = ceiling((r+g+b)/2)) within each fixed odd (r+g+b) layer is the number of "fighting fish" with fixed numbers of left lower free and right lower free edges with a marked tail A278880.

Links

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

T. Einolf, R. Muth and J. Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021, Remark 4.7.

Formula

T(r,g,b) = (r+g+b)/((g+b)*(r+b))*C(r+g,b-1)*C(g+b,r)*C(r+b,g).

T(r,g,b) = (r+g+b)/((g+b)*(r+b))*(r+g)!/((r+g-b+1)!*(b-1)!)*((g+b)!/(g+b-r)!*r!))*((r+b)!/((r+b-g)!*g!).

Example

The first few layers of the pyramidal array are:
-----------------------------------------------------------------------
      1           (r+g+b=1), (b=1)           T(0,0,1)
                                                         LAYER SUM:   1
-----------------------------------------------------------------------
     1 1          (r+g+b=2), (b=1)        T(0,1,1) T(1,0,1)
                                                         LAYER SUM:   2
-----------------------------------------------------------------------
      3           (r+g+b=3), (b=1)           T(1,1,1)
     1 1          (r+g+b=3), (b=2)        T(0,1,2) T(1,0,2)
                                                         LAYER SUM:   5
-----------------------------------------------------------------------
     2 2          (r+g+b=4), (b=1)        T(1,2,1) T(2,1,1)
    1 8 1         (r+g+b=4), (b=2)     T(0,2,2) T(1,1,2) T(2,0,2)
                                                         LAYER SUM:  14
-----------------------------------------------------------------------
      5           (r+g+b=5), (b=1)           T(2,2,1)
    15 15         (r+g+b=5), (b=2)        T(1,2,2) T(2,1,2)
   1  5  1        (r+g+b=5), (b=3)     T(0,2,3) T(1,1,3) T(2,0,3)
                                                         LAYER SUM:  42
-----------------------------------------------------------------------
     3  3         (r+g+b=6), (b=1)        T(2,3,1) T(3,2,1)
   8  54  8       (r+g+b=6), (b=2)     T(1,3,2) T(2,2,2) T(3,1,2)
 1  27  27  1     (r+g+b=6), (b=3)  T(0,3,3) T(1,2,3) T(2,1,3) T(3,0,3)
                                                         LAYER SUM: 132
-----------------------------------------------------------------------
      7           (r+g+b=7), (b=1)            T(3,3,1)
   70   70        (r+g+b=7), (b=2)        T(2,3,2) T(3,2,2)
 42  168  42      (r+g+b=7), (b=3)     T(1,3,3) T(2,2,3) T(3,1,3)
1   14  14   1    (r+g+b=7), (b=4)  T(0,3,4) T(1,2,4) T(2,1,4) T(3,0,4)
                                                         LAYER SUM: 429
-----------------------------------------------------------------------

Crossrefs

Cf. A000108, A108759, A278880.

Sequence in context: A301330 A175636 A352935 * A204253 A064048 A336218

Adjacent sequences: A364754 A364755 A364756 * A364758 A364759 A364760

Keyword

nonn,tabf

Author

Robert Muth, Aug 05 2023

Status

approved