cp's OEIS Frontend

A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

%I A380362 #10 Jan 26 2025 17:42:37
%S A380362 1,0,1,0,1,1,0,0,1,1,0,0,1,2,1,0,0,0,3,2,1,0,0,0,3,6,3,1,0,0,0,0,7,11,
%T A380362 3,1,0,0,0,0,4,24,17,4,1,0,0,0,0,0,24,51,26,4,1,0,0,0,0,0,12,89,109,
%U A380362 36,5,1,0,0,0,0,0,0,74,265,194,50,5,1,0,0,0,0,0,0,27,371,660,345,65,6,1
%N A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.
%C A380362 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 A380362 The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.
%C A380362 T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotations and reflections.
%H A380362 Andrew Howroyd, <a href="/A380362/b380362.txt">Table of n, a(n) for n = 4..1278</a> (first 50 rows)
%H A380362 Andrew Howroyd, <a href="/A380362/a380362.txt">Formulas and PARI Program</a>, Jan 2025.
%H A380362 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalinGraph.html">Halin Graph</a>.
%H A380362 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circuit_rank">Circuit rank</a>.
%H A380362 Wikipedia, <a href="https://en.wikipedia.org/wiki/Halin_graph">Halin graph</a>.
%F A380362 T(n,k) = A295634(k, n-k).
%e A380362 Triangle begins:
%e A380362   n\k| 3  4  5  6  7   8   9   10  11  12  13
%e A380362 -----+----------------------------------------
%e A380362    4 | 1;
%e A380362    5 | 0, 1;
%e A380362    6 | 0, 1, 1;
%e A380362    7 | 0, 0, 1, 1;
%e A380362    8 | 0, 0, 1, 2, 1;
%e A380362    9 | 0, 0, 0, 3, 2,  1;
%e A380362   10 | 0, 0, 0, 3, 6,  3,  1;
%e A380362   11 | 0, 0, 0, 0, 7, 11,  3,   1;
%e A380362   12 | 0, 0, 0, 0, 4, 24, 17,   4,  1;
%e A380362   13 | 0, 0, 0, 0, 0, 24, 51,  26,  4,  1;
%e A380362   14 | 0, 0, 0, 0, 0, 12, 89, 109, 36,  5,  1;
%e A380362    ...
%o A380362 (PARI) \\ See PARI Link for program code.
%o A380362 { my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }
%Y A380362 Row sums are A346779.
%Y A380362 Column sums are A001004.
%Y A380362 Main diagonal is A000012.
%Y A380362 Central coefficients are A000207.
%Y A380362 Cf. A295634, A380361.
%K A380362 nonn,tabl
%O A380362 4,14
%A A380362 _Andrew Howroyd_, Jan 25 2025