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A380361 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.

%I A380361 #12 Jan 26 2025 17:42:42
%S A380361 1,0,1,0,1,1,0,0,1,1,0,0,1,2,1,0,0,0,4,2,1,0,0,0,4,8,3,1,0,0,0,0,12,
%T A380361 16,3,1,0,0,0,0,6,40,25,4,1,0,0,0,0,0,43,93,40,4,1,0,0,0,0,0,19,165,
%U A380361 197,56,5,1,0,0,0,0,0,0,143,505,364,80,5,1
%N A380361 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.
%C A380361 The circuit rank is equal to the number of leaves on the tree before it is extended into a Halin graph by joining up the leaves.
%C A380361 The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.
%C A380361 T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotation.
%H A380361 Andrew Howroyd, <a href="/A380361/b380361.txt">Table of n, a(n) for n = 4..1278</a> (first 50 rows)
%H A380361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalinGraph.html">Halin Graph</a>.
%H A380361 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circuit_rank">Circuit rank</a>.
%H A380361 Wikipedia, <a href="https://en.wikipedia.org/wiki/Halin_graph">Halin graph</a>.
%F A380361 T(n,k) = A295633(k, n-k).
%e A380361 Triangle begins:
%e A380361   n\k| 3  4  5  6   7   8    9   10  11  12  13
%e A380361 -----+-----------------------------------------
%e A380361    4 | 1;
%e A380361    5 | 0, 1;
%e A380361    6 | 0, 1, 1;
%e A380361    7 | 0, 0, 1, 1;
%e A380361    8 | 0, 0, 1, 2,  1;
%e A380361    9 | 0, 0, 0, 4,  2,  1;
%e A380361   10 | 0, 0, 0, 4,  8,  3,   1;
%e A380361   11 | 0, 0, 0, 0, 12, 16,   3,   1;
%e A380361   12 | 0, 0, 0, 0,  6, 40,  25,   4,  1;
%e A380361   13 | 0, 0, 0, 0,  0, 43,  93,  40,  4,  1;
%e A380361   14 | 0, 0, 0, 0,  0, 19, 165, 197, 56,  5,  1;
%e A380361   ...
%o A380361 (PARI) \\ See PARI Link in A380362 for program code.
%o A380361 { my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }
%Y A380361 Row sums are A380360.
%Y A380361 Column sums are A003455.
%Y A380361 Main diagonal is A000012.
%Y A380361 Central coefficients are A001683.
%Y A380361 Cf. A295633, A380362.
%K A380361 nonn,tabl
%O A380361 4,14
%A A380361 _Andrew Howroyd_, Jan 25 2025