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 A001854 Total height of all rooted trees on n labeled nodes. (Formerly M2081 N0822) 8
 0, 2, 15, 148, 1785, 26106, 449701, 8927192, 200847681, 5053782070, 140679853941, 4293235236324, 142553671807729, 5116962926162738, 197459475792232725, 8152354312656732976, 358585728464893234305, 16741214317684425260142, 826842457727306803110997, 43073414675338753123113980 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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 REFERENCES Rényi, A., and G. Szekeres. "On the height of trees." Journal of the Australian Mathematical Society 7.04 (1967): 497-507. See (4.7). N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..387 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478. J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy) FORMULA a(n) = Sum_{k=1..n-1} A034855(n,k)*k. - Geoffrey Critzer, Mar 14 2013 A000435(n)/a(n) ~ 1/2 (see A000435 and the Renyi-Szekeres result mentioned in the Comments). - David desJardins, Jan 20 2017 MATHEMATICA 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 *) CROSSREFS Cf. A000435, A034855, A236396. Also A234953(n) = a(n)/n. Sequence in context: A224885 A253571 A111686 * A060226 A325060 A002103 Adjacent sequences:  A001851 A001852 A001853 * A001855 A001856 A001857 KEYWORD nonn AUTHOR EXTENSIONS More terms from Geoffrey Critzer, Mar 14 2013 STATUS approved

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Last modified October 20 04:37 EDT 2019. Contains 328247 sequences. (Running on oeis4.)