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A356186
Number of labeled trees on [2n] with a bicentroid.
1
0, 1, 12, 810, 143360, 49218750, 27935373312, 23751648836916, 28301429298954240, 45046920790988254710, 92378000000000000000000, 237289687212632836205339916, 746430126201849206626773368832, 2822726846177838977566127355808300
OFFSET
0,3
COMMENTS
This sequence is the labeled version of A102911 where the pertinent definitions can be found.
FORMULA
a(n) = binomial(2n,n)*n^(2n-2)/2 = A000984(n)*A000169(n)^2/2.
EXAMPLE
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.
MATHEMATICA
Prepend[Table[Binomial[2 n, n] n^(n - 1) n^(n - 1)/2, {n, 1, 12}], 0]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jul 31 2022
STATUS
approved