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

Author

Robert Muth, Aug 05 2023

Keywords

• nonn
• tabf

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.

Examples

			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
-----------------------------------------------------------------------
		

Links

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

Crossrefs

Cf. A000108, A108759, A278880.

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!)).