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%I #46 Jul 31 2022 19:52:48
%S 0,1,12,810,143360,49218750,27935373312,23751648836916,
%T 28301429298954240,45046920790988254710,92378000000000000000000,
%U 237289687212632836205339916,746430126201849206626773368832,2822726846177838977566127355808300
%N Number of labeled trees on [2n] with a bicentroid.
%C This sequence is the labeled version of A102911 where the pertinent definitions can be found.
%H N. J. A. Sloane, <a href="/A000055/a000055.gif">Illustration of initial terms</a>
%F a(n) = binomial(2n,n)*n^(2n-2)/2 = A000984(n)*A000169(n)^2/2.
%e a(3) = 810. In the illustrations by Sloane found in the link above, for n = 6, there are A102911(3) = 3 trees with a bicentroid: the first, second and last trees shown. They have 360, 360, and 90 labelings respectively. 360 + 360 + 90 = 810.
%t Prepend[Table[Binomial[2 n, n] n^(n - 1) n^(n - 1)/2, {n, 1, 12}], 0]
%Y Cf. A102911, A000984, A000169.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Jul 31 2022
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