cp's OEIS Frontend

A381486 Number of labeled histories for rooted ternary trees with 2n+1 leaves if simultaneous trifurcations are allowed.

%I A381486 #15 Feb 26 2025 09:51:30
%S A381486 1,1,10,420,43960,9347800,3513910400,2131249120000,1952028782704000,
%T A381486 2568150610833808000,4666919676058159520000,
%U A381486 11351087418588355518080000,36008099327884173922432000000,145785514242304854141480256000000,739598808823839440680777500928000000,4627885522642342503645368137231360000000
%N A381486 Number of labeled histories for rooted ternary trees with 2n+1 leaves if simultaneous trifurcations are allowed.
%C A381486 a(n) is also the number of single-elimination sports tournament schedules possible for 2n+1 teams if matches involve three teams, arbitrarily many arenas are available, and labeled teams have been specified, but the bracket of matches has not been specified.
%H A381486 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.
%F A381486 a(n) = Y(2n+1), where Y(n) = Sum_{i=1..floor(n/3)} (n!/(i!*6^i*(n-3*i)!))*Y(n-2*i), with Y(1)=1.
%e A381486 Consider 7 named players in a sport in which players compete 3 at a time (e.g. the television gameshow "Jeopardy!"). The number of ways a single-elimination tournament can be arranged, if simultaneous matches can take place, is a(3)=420. Three of these 420 are: (1) A, B, and C play; the winner plays against D and E; the winner plays against F and G. (2) D, E, and F play; the winner plays against A and B; the winner plays against C and G. (3) A, B, and C play simultaneous with D, E, and F; the winners of these matches play against G.
%p A381486 a:= proc(n) option remember; `if`(n=0, 1, add((2*n+1)!/
%p A381486       (i!*6^i*(2*n+1-3*i)!)*a(n-i), i=1..(2*n+1)/3))
%p A381486     end:
%p A381486 seq(a(n), n=0..15);  # _Alois P. Heinz_, Feb 25 2025
%Y A381486 Cf. A317059 for binary rather than ternary trees, A339411 if simultaneity is disallowed.
%K A381486 nonn
%O A381486 0,3
%A A381486 _Noah A Rosenberg_, Feb 25 2025