This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A364757 #20 Apr 13 2025 07:11:34 %S A364757 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 A364757 168,42,1,14,14,1,4,4,30,192,30,20,400,400,20,1,64,200,64,1,9,210,210, %U A364757 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 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. %C A364757 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 A364757 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 A364757 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 A364757 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 A364757 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 A364757 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 A364757 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 A364757 The first few layers of the pyramidal array are: %e A364757 ----------------------------------------------------------------------- %e A364757 1 (r+g+b=1), (b=1) T(0,0,1) %e A364757 LAYER SUM: 1 %e A364757 ----------------------------------------------------------------------- %e A364757 1 1 (r+g+b=2), (b=1) T(0,1,1) T(1,0,1) %e A364757 LAYER SUM: 2 %e A364757 ----------------------------------------------------------------------- %e A364757 3 (r+g+b=3), (b=1) T(1,1,1) %e A364757 1 1 (r+g+b=3), (b=2) T(0,1,2) T(1,0,2) %e A364757 LAYER SUM: 5 %e A364757 ----------------------------------------------------------------------- %e A364757 2 2 (r+g+b=4), (b=1) T(1,2,1) T(2,1,1) %e A364757 1 8 1 (r+g+b=4), (b=2) T(0,2,2) T(1,1,2) T(2,0,2) %e A364757 LAYER SUM: 14 %e A364757 ----------------------------------------------------------------------- %e A364757 5 (r+g+b=5), (b=1) T(2,2,1) %e A364757 15 15 (r+g+b=5), (b=2) T(1,2,2) T(2,1,2) %e A364757 1 5 1 (r+g+b=5), (b=3) T(0,2,3) T(1,1,3) T(2,0,3) %e A364757 LAYER SUM: 42 %e A364757 ----------------------------------------------------------------------- %e A364757 3 3 (r+g+b=6), (b=1) T(2,3,1) T(3,2,1) %e A364757 8 54 8 (r+g+b=6), (b=2) T(1,3,2) T(2,2,2) T(3,1,2) %e A364757 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 A364757 LAYER SUM: 132 %e A364757 ----------------------------------------------------------------------- %e A364757 7 (r+g+b=7), (b=1) T(3,3,1) %e A364757 70 70 (r+g+b=7), (b=2) T(2,3,2) T(3,2,2) %e A364757 42 168 42 (r+g+b=7), (b=3) T(1,3,3) T(2,2,3) T(3,1,3) %e A364757 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 A364757 LAYER SUM: 429 %e A364757 ----------------------------------------------------------------------- %Y A364757 Cf. A000108, A108759, A278880. %K A364757 nonn,tabf %O A364757 1,4 %A A364757 _Robert Muth_, Aug 05 2023