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A294522 Triangle read by rows: T(n,k) is the number of simple connected graphs on n nodes with diameter k (0<=k

%I A294522 #17 Feb 16 2025 08:33:51
%S A294522 1,0,1,0,1,1,0,1,4,1,0,1,14,5,1,0,1,59,43,8,1,0,1,373,387,82,9,1,0,1,
%T A294522 4154,5797,1027,125,12,1,0,1,91518,148229,19320,1818,180,13,1,0,1,
%U A294522 4116896,6959721,598913,37856,2928,239,16,1
%N A294522 Triangle read by rows: T(n,k) is the number of simple connected graphs on n nodes with diameter k (0<=k<n).
%H A294522 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDiameter.html">Graph Diameter</a>
%H A294522 Wikipedia, <a href="https://en.wikipedia.org/wiki/Distance_(graph_theory)">Distance (graph theory)</a>
%F A294522 a(n,1) = 1 for n > 1 (only K_n has diameter 1).
%F A294522 a(n,n-1) = 1 (only P_n has diameter n-1).
%e A294522 Triangle begins:
%e A294522   1;
%e A294522   0, 1;
%e A294522   0, 1,    1;
%e A294522   0, 1,    4,    1;
%e A294522   0, 1,   14,    5,    1;
%e A294522   0, 1,   59,   43,    8,   1;
%e A294522   0, 1,  373,  387,   82,   9,  1;
%e A294522   0, 1, 4154, 5797, 1027, 125, 12, 1;
%e A294522   ...
%e A294522 From _Eric W. Weisstein_, Jun 11 2019: (Start)
%e A294522 a(2,1) = 1 since only P_2 has diameter 1.
%e A294522 a(3,1) = 1 since only C_3 has diameter 1.
%e A294522 a(3,2) = 1 since only P_3 has diameter 2.
%e A294522 a(4,1) = 1 since only K_4 has diameter 1.
%e A294522 a(4,2) = 4 since K_1,3, K4-e, the paw graph, and C_4 have diameter 2.
%e A294522 a(4,3) = 1 since only P_4 has diameter 3.
%e A294522 (End)
%Y A294522 Columns k=0..6 are A000007, A057427, A241706, A241707, A241708, A241709, A241710.
%Y A294522 Row sums give A001349.
%K A294522 nonn,tabl
%O A294522 1,9
%A A294522 _Andrew Howroyd_, Nov 01 2017