A356186 Number of labeled trees on [2n] with a bicentroid.

0, 1, 12, 810, 143360, 49218750, 27935373312, 23751648836916, 28301429298954240, 45046920790988254710, 92378000000000000000000, 237289687212632836205339916, 746430126201849206626773368832, 2822726846177838977566127355808300

Offset: 0

Author

Geoffrey Critzer, Jul 31 2022

Keywords

• nonn

Comments

This sequence is the labeled version of A102911 where the pertinent definitions can be found.

Examples

			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.
		

Links

• N. J. A. Sloane, Illustration of initial terms

Crossrefs

Cf. A102911, A000984, A000169.

Programs

Mathematica

Prepend[Table[Binomial[2 n, n] n^(n - 1) n^(n - 1)/2, {n, 1, 12}], 0]

Formula

a(n) = binomial(2n,n)*n^(2n-2)/2 = A000984(n)*A000169(n)^2/2.