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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
 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 (list; graph; refs; listen; history; text; internal format)
 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: A364753 A364755 A364756 * A364758 A364759 A364760 KEYWORD nonn,tabf AUTHOR Robert Muth, Aug 05 2023 STATUS approved

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Last modified November 29 07:03 EST 2023. Contains 367429 sequences. (Running on oeis4.)