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A064060 Number of connected, homeomorphically irreducible (also called series-reduced) trees with n >= 2 labeled leaves (numbers in nondecreasing order).

%I A064060 #20 Aug 27 2019 12:14:03
%S A064060 1,1,1,3,1,10,15,1,10,15,15,45,60,90,1,21,35,70,105,105,105,105,210,
%T A064060 315,420,630,630,1,28,35,56,105,168,210,210,280,280,280,315,420,420,
%U A064060 560,560,840,840,840,1260,1260
%N A064060 Number of connected, homeomorphically irreducible (also called series-reduced) trees with n >= 2 labeled leaves (numbers in nondecreasing order).
%C A064060 The number of entries of row n of this array is A007827(n), n >= 2.
%C A064060 With v the total number of nodes (vertices), e the number of edges (links), n >= 2 the number of edges ending in a degree 1 node (leaves), i the number of edges which end in nodes with degree >=3 (internal edges) and v_{d} the number of nodes of degree d=1,3,4,... one has: v = e+1 = n + Sum_{d>=3}v_{d}, i = e-n, Sum_{d>=3}d*v_{d} = 2(v-1)-n.
%H A064060 Ch. Mayer, <a href="/A064060/a064060.ps">Illustration</a>
%H A064060 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%e A064060 Irregular array starts:
%e A064060 {1};
%e A064060 {1};
%e A064060 {1, 3};
%e A064060 {1, 10, 15};
%e A064060 {1, 10, 15, 15, 15, 45, 60, 90};
%e A064060 {1, 21, 35, 70, 105, 105, 105, 105, 210, 315, 420, 630, 630};
%e A064060 {1, 28, 35, 56, 105, 168, 210, 210, 280, 280, 280, 315, 420, 420, 560, 560, 840, 840, 840, 1260, 1260, 1680, 1680, 1680, 1680, 2520, 2520, 2520, 2520, 3360, 5040, 5040};
%e A064060 ...
%Y A064060 The row sums give A000311(n-1), n >= 2. Cf. A007827.
%K A064060 nonn,tabf
%O A064060 2,4
%A A064060 _Wolfdieter Lang_ and Christoph Mayer (Christoph.Mayer(AT)dlr.de), Sep 13 2001