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 A381865 #9 Mar 19 2025 10:25:12 %S A381865 1,1,13,308682013,20447648974223714249697186722386536049691073 %N A381865 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 3^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 3 players. %C A381865 a(n) is also the number of tie-permitting labeled histories for a fully symmetric strictly trifurcating labeled topology with 3^n leaves. %H A381865 Emily H. Dickey and Noah A. Rosenberg, <a href="https://doi.org/10.1098/rstb.2023.0307">Labelled histories with multifurcation and simultaneity</a>, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Theorem 15 with r=3) %e A381865 Two of the 13 cases with n=2 and 3^2=9 players are: (1) (A,B,C) play, then (D,E,F) play, then (G,H,I) play, then the winners of the three matches play; (2) (A,B,C) play simultaneously with (D,E,F), then the winners of these two matches play against G, then the winner plays against H and I. %Y A381865 Cf. A273723 (if matches must be non-simultaneous), A379758 (if matches involve only two players at a time). %K A381865 nonn,more %O A381865 0,3 %A A381865 _Noah A Rosenberg_, Mar 08 2025