This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356073 #15 Aug 04 2022 15:54:36
%S A356073 0,1,0,3,4,125,486,16807,118784,4782969,50781250,2357947691,
%T A356073 33981990912,1792160394037,32942263538380,1946195068359375,
%U A356073 43756164738973696,2862423051509815793
%N A356073 Number of labeled trees on [n] having a centroid.
%C A356073 This is the labeled version of A027416 where the pertinent definitions can be found.
%H A356073 N. J. A. Sloane, <a href="/A000055/a000055.gif">Illustration of initial terms</a>
%F A356073 If n is even, a(n) = n^(n-2) - binomial(n,n/2)*(n/2)^(n-2)/2 = A000272(n) - A000984(n)*A356186(n/2)/2.
%F A356073 If n is odd, a(n) = n^(n-2).
%F A356073 For n >= 1, a(n) = n!*[x^n] x*exp(T(x)) where T(x) = Sum_{i=1..ceiling(n/2)-1} i^(i-1)x^i/i!.
%e A356073 a(6) = 486. In the illustrations by Sloane found in the link above, for n = 6, there are A027416(3) = 3 trees with a centroid: the third, fourth and fifth trees shown. They have 360, 120, and 6 labelings respectively. 360 + 120 + 6 = 486.
%t A356073 Prepend[Table[If[EvenQ[n], n^(n - 2) - Binomial[n, n/2] (n/2)^(n - 2)/2, n^(n - 2)], {n, 1, 17}], 0]
%Y A356073 Cf. A356186, A000272, A027416.
%K A356073 nonn
%O A356073 0,4
%A A356073 _Geoffrey Critzer_, Jul 31 2022