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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. 0

%I #16 Sep 06 2023 21:02:20

%S 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,

%T 168,42,1,14,14,1,4,4,30,192,30,20,400,400,20,1,64,200,64,1,9,210,210,

%U 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

%N 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.

%C 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.

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

%C 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).

%C 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.

%H T. Einolf, R. Muth and J. Wilkinson, <a href="https://arxiv.org/abs/2107.13417">Injectively k-colored rooted forests</a>, arXiv:2107.13417 [math.CO], 2021, Remark 4.7.

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

%F 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!).

%e The first few layers of the pyramidal array are:

%e -----------------------------------------------------------------------

%e 1 (r+g+b=1), (b=1) T(0,0,1)

%e LAYER SUM: 1

%e -----------------------------------------------------------------------

%e 1 1 (r+g+b=2), (b=1) T(0,1,1) T(1,0,1)

%e LAYER SUM: 2

%e -----------------------------------------------------------------------

%e 3 (r+g+b=3), (b=1) T(1,1,1)

%e 1 1 (r+g+b=3), (b=2) T(0,1,2) T(1,0,2)

%e LAYER SUM: 5

%e -----------------------------------------------------------------------

%e 2 2 (r+g+b=4), (b=1) T(1,2,1) T(2,1,1)

%e 1 8 1 (r+g+b=4), (b=2) T(0,2,2) T(1,1,2) T(2,0,2)

%e LAYER SUM: 14

%e -----------------------------------------------------------------------

%e 5 (r+g+b=5), (b=1) T(2,2,1)

%e 15 15 (r+g+b=5), (b=2) T(1,2,2) T(2,1,2)

%e 1 5 1 (r+g+b=5), (b=3) T(0,2,3) T(1,1,3) T(2,0,3)

%e LAYER SUM: 42

%e -----------------------------------------------------------------------

%e 3 3 (r+g+b=6), (b=1) T(2,3,1) T(3,2,1)

%e 8 54 8 (r+g+b=6), (b=2) T(1,3,2) T(2,2,2) T(3,1,2)

%e 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)

%e LAYER SUM: 132

%e -----------------------------------------------------------------------

%e 7 (r+g+b=7), (b=1) T(3,3,1)

%e 70 70 (r+g+b=7), (b=2) T(2,3,2) T(3,2,2)

%e 42 168 42 (r+g+b=7), (b=3) T(1,3,3) T(2,2,3) T(3,1,3)

%e 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)

%e LAYER SUM: 429

%e -----------------------------------------------------------------------

%Y Cf. A000108, A108759, A278880.

%K nonn,tabf

%O 1,4

%A _Robert Muth_, Aug 05 2023

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