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A383448 Irregular triangle read by rows: T(n,k) (n>=1, k>=0) is the number of trees with n nodes in which there are k edges whose end-vertices have the same degree.

%I A383448 #31 May 04 2025 12:48:53
%S A383448 1,0,1,1,1,1,2,0,1,3,2,0,1,6,3,1,0,1,9,7,5,1,0,1,19,12,10,4,1,0,1,33,
%T A383448 33,18,15,5,1,0,1,67,66,54,26,15,5,1,0,1,130,154,128,77,36,18,6,1,0,1,
%U A383448 270,344,309,199,110,40,21,6,1,0,1,547,806,752,530,294,147,50,24,7,1,0,1
%N A383448 Irregular triangle read by rows: T(n,k) (n>=1, k>=0) is the number of trees with n nodes in which there are k edges whose end-vertices have the same degree.
%C A383448 T(n,0) is the number of "peerless" trees on n nodes (see A383447). The row sums are A000055.
%C A383448 For n >= 3, row n has n-2 entries.
%D A383448 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 233.
%H A383448 N. J. A. Sloane, <a href="/A383448/a383448.jpg">Illustration for row 8</a> (The 23 trees with 8 nodes are numbered in black ink in the order in which they appear in Harary's table. Edges with equal degree nodes are drawn in red and their number is shown in red ink.)
%H A383448 Jakub Buczak, <a href="/A383448/a383448.txt">Rows 1 to 20 of the triangle, flattened.</a>
%e A383448 Triangle begins:
%e A383448    1,
%e A383448    0, 1,
%e A383448    1,
%e A383448    1, 1,
%e A383448    2, 0, 1,
%e A383448    3, 2, 0, 1,
%e A383448    6, 3, 1, 0, 1,
%e A383448    9, 7, 5, 1, 0, 1,
%e A383448    19, 12, 10, 4, 1, 0, 1,
%e A383448    33, 33, 18, 15, 5, 1, 0, 1,
%e A383448    67, 66, 54, 26, 15, 5, 1, 0, 1,
%e A383448    130, 154, 128, 77, 36, 18, 6, 1, 0, 1,
%e A383448    270, 344, 309, 199, 110, 40, 21, 6, 1, 0, 1,
%e A383448    547, 806, 752, 530, 294, 147, 50, 24, 7, 1, 0, 1
%e A383448    ...
%e A383448 Enough rows are shown to demonstrate that the leading entry only dominates for small n. - _N. J. A. Sloane_, May 04 2025
%Y A383448 Cf. A000055, A383447.
%K A383448 nonn,tabf
%O A383448 1,7
%A A383448 _N. J. A. Sloane_, May 01 2025
%E A383448 More terms from _Jakub Buczak_, May 04 2025.