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%I #23 Jan 15 2025 08:47:59
%S 1,3,365,1323338487,1119556146543237253601352961,
%T 3414445659328795239581367793706562556567987857578516541118092297328702035
%N Number of sequences in which the games of a fully symmetric single-elimination tournament with 2^n teams can be played if arbitrarily many arenas are available.
%C a(n) is also the number of tie-permitting labeled histories for a fully symmetric labeled topology with 2^n leaves.
%H M. C. King and N. 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 a(n) = Sum_{k=n..2^n-1} A380166(n,k).
%e For n=2 and a tournament with structure ((A,B),(C,D)), game (A,B) can be played before, after, or simultaneously with game (C,D), producing a(2)=3.
%Y Cf. A056972 (game sequences with only one arena).
%Y a(n) gives row sums for A380166(n,k).
%K nonn
%O 1,2
%A _Noah A Rosenberg_, Jan 01 2025