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 A001854 M2081 N0822 #47 Jun 21 2022 19:56:06
%S A001854 0,2,15,148,1785,26106,449701,8927192,200847681,5053782070,
%T A001854 140679853941,4293235236324,142553671807729,5116962926162738,
%U A001854 197459475792232725,8152354312656732976,358585728464893234305,16741214317684425260142,826842457727306803110997,43073414675338753123113980
%N A001854 Total height of all rooted trees on n labeled nodes.
%C A001854 Take any one of the n^(n-1) rooted trees on n labeled nodes, compute its height (maximal edge distance to root), sum over all trees.
%C A001854 Theorem [Renyi-Szekeres, (4,7)]. The average height if the tree is chosen at random is sqrt(2*n*Pi). - _David desJardins_, Jan 20 2017
%D A001854 Rényi, A., and G. Szekeres. "On the height of trees." Journal of the Australian Mathematical Society 7.04 (1967): 497-507. See (4.7).
%D A001854 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001854 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001854 Alois P. Heinz, <a href="/A001854/b001854.txt">Table of n, a(n) for n = 1..387</a>
%H A001854 J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%H A001854 J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
%H A001854 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A001854 a(n) = Sum_{k=1..n-1} A034855(n,k)*k. - _Geoffrey Critzer_, Mar 14 2013
%F A001854 A000435(n)/a(n) ~ 1/2 (see A000435 and the Renyi-Szekeres result mentioned in the Comments). - _David desJardins_, Jan 20 2017
%t A001854 nn=20;a=NestList[ x Exp[#]&,x,nn];f[list_]:=Sum[list[[i]]*i,{i,1,Length[list]}];Drop[Map[f,Transpose[Table[Range[0,nn]!CoefficientList[Series[a[[i+1]]-a[[i]],{x,0,nn}],x],{i,1,nn-1}]]],1] (* _Geoffrey Critzer_, Mar 14 2013 *)
%Y A001854 Cf. A000435, A034855, A236396.
%Y A001854 Also A234953(n) = a(n)/n.
%K A001854 nonn
%O A001854 1,2
%A A001854 _N. J. A. Sloane_
%E A001854 More terms from _Geoffrey Critzer_, Mar 14 2013