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%I #16 Jul 03 2022 12:04:05
%S 1,1,1,1,2,1,2,3,4,3,2,6,7,10,8,5,8,18,19,29,23,5,18,26,52,57,86,68,
%T 14,23,68,82,166,176,266,215,14,56,91,220,270,524,557,844,680,42,70,
%U 248,321,769,890,1722,1806,2742,2226,42,180,318,872,1151,2568,2986,5664,5954,9032,7327
%N Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane that have reflection symmetry, n >= 0, k >= 0.
%C "A closed bounded region in the plane divided into triangular regions with k+3 vertices on the boundary and n internal vertices is said to be a triangular map of type [n,k]." It is a [n,k]-triangulation if there are no multiple edges.
%C "... may be evaluated from the results given by Brown."
%D C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
%H Andrew Howroyd, <a href="/A169809/b169809.txt">Table of n, a(n) for n = 0..1325</a>
%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%H C. F. Earl and L. J. March, <a href="/A005500/a005500_1.pdf">Architectural applications of graph theory</a>, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
%e Array begins:
%e ====================================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+-----------------------------------------------
%e 0 | 1 1 1 2 2 5 5 14 ...
%e 1 | 1 2 3 6 8 18 23 56 ...
%e 2 | 1 4 7 18 26 68 91 248 ...
%e 3 | 3 10 19 52 82 220 321 872 ...
%e 4 | 8 29 57 166 270 769 1151 3296 ...
%e 5 | 23 86 176 524 890 2568 4020 11558 ...
%e 6 | 68 266 557 1722 2986 8902 14197 42026 ...
%e 7 | 215 844 1806 5664 10076 30362 49762 148208 ...
%e ...
%o (PARI) \\ See link in A169808 for script.
%o A169809Array(7) \\ _Andrew Howroyd_, Feb 22 2021
%Y Columns k=0..3 are A002712, A005505, A005506, A005507.
%Y Rows n=0..2 are A208355, A005508, A005509.
%Y Antidiagonal sums give A005028.
%Y Cf. A146305 (rooted), A169808 (unrooted), A262586 (oriented).
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, May 25 2010
%E Edited and terms a(36) and beyond from _Andrew Howroyd_, Feb 22 2021
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