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 A378855 #22 Mar 07 2025 14:13:31 %S A378855 1,0,1,0,1,2,0,0,2,3,0,0,2,9,8,0,0,1,12,30,20,0,0,1,22,102,160,80,0,0, %T A378855 0,10,114,380,485,210,0,0,0,10,198,1100,2495,2478,896,0,0,0,5,204, %U A378855 1930,7260,12810,10640,3360,0,0,0,5,344,4890,27110,72702 %N A378855 Triangle read by rows: T(n,k) is the number of sequences in which the games of a single-elimination tournament with n teams can be played if arbitrarily many arenas are available, the tournament bracket is chosen to the bracket with the largest such number of sequences, and the number of distinct times at which games are played is k, log_2(n) <= k <= n-1. %C A378855 T(n,k) is also the number of tie-permitting labeled histories for a labeled topology with n leaves and exactly k times at which events take place, when the labeled topology is chosen to be the labeled topology with the largest number of tie-permitting labeled histories across all labeled topologies with n leaves. %C A378855 The first row has n=2. Terms for n=2 to 8 appear in Tables 2 and 3 of King & Rosenberg (2023); terms for n=9 to 16 are supplied by Emily H. Dickey. %H A378855 Matthew C. King and Noah A. Rosenberg, <a href="https://doi.org/10.1080/0025570X.2023.2266389">A mathematical connection between single-elimination sports tournaments and evolutionary trees</a>, Math. Mag. 96 (2023), 484-497. %F A378855 The maximum is computed over unlabeled binary rooted trees T with n leaves (trees in the set enumerated by A001190) of the quantity computed for tree T in eq. 3 of King & Rosenberg (2023). This maximum gives the row sum, tabulated in A380767. For the tree that generates the maximum, the row entries are computed as the specific terms described in Theorem 3 of King & Rosenberg (2023) (and summed in eq. 3). %e A378855 Triangle begins: %e A378855 1; %e A378855 0, 1; %e A378855 0, 1, 2; %e A378855 0, 0, 2, 3; %e A378855 0, 0, 2, 9, 8; %e A378855 0, 0, 1, 12, 30, 20; %e A378855 0, 0, 1, 22, 102, 160, 80; %e A378855 0, 0, 0, 10, 114, 380, 485, 210; %e A378855 0, 0, 0, 10, 198, 1100, 2495, 2478, 896; %e A378855 0, 0, 0, 5, 204, 1930, 7260, 12810, 10640, 3360; %e A378855 0, 0, 0, 5, 344, 4890, 27110, 72702, 101024, 70080, 19200; %e A378855 0, 0, 0, 2, 278, 6360, 53000, 211365, 451164, 529116, 321600, 79200; %Y A378855 Row sums are A380767. %Y A378855 Cf. A380166 for the triangle if n is a power of 2. %Y A378855 Entries T(n,n-1) follow A056971. %K A378855 nonn,tabl %O A378855 2,6 %A A378855 _Noah A Rosenberg_, Feb 10 2025