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A060972 Triangle giving numbers of closed plane meanders.

%I A060972 #40 Dec 15 2015 19:10:55
%S A060972 1,1,1,1,6,1,1,20,20,1,1,50,160,50,1,1,105,808,808,105,1,1,196,3066,
%T A060972 7294,3066,196,1,1,336,9552,45588,45588,9552,336,1,1,540,25740,220362,
%U A060972 440172,220362,25740,540,1,1,825,62040,879840,3133724,3133724,879840,62040,825,1,1,1210,136851,3028454,17752636,31586346,17752636,3028454,136851,1210,1
%N A060972 Triangle giving numbers of closed plane meanders.
%C A060972 a(n) counts closed plane meanders according to the number of white regions when regions are colored black and white alternatively. So the sum of each row is given by A005315. The outer columns consist of 1's. The next-to-outer columns are given by A002415.
%C A060972 This is also the number of arches above the x-axis going from an odd vertex to a higher even vertex(p) for closed plane meanders(M) with n arches. By symmetry, these same subsets exist for arches below the x-axis. For each meander solution, the total arches for the top and bottom that go from an odd vertex to a higher even vertex is n+1.
%C A060972 Example: M(n,p): M(3,1)=1 [(top 16,23,45; bottom 12,34,56)], M(3,2)=6 [(top 14,23,56; bottom 16,25,34)(top 16,25,34; bottom 14,23,56) (top 12,36,45; bottom 16,25,34) (top 16,25,34; bottom 12,36,45) (top 12,36,45; bottom 14,23,56)(top 14,23,56; bottom 12,36,45)] M(3,3)=1 [(top 12,34,56; bottom 16,23,45)]. - _Roger Ford_, Sep 29 2014
%H A060972 Joerg Arndt and Andrew Howroyd, <a href="/A060972/b060972.txt">Table of n, a(n) for n = 1..210</a> (first 105 terms from Joerg Arndt)
%H A060972 Reinhard O. W. Franz, and Berton A. Earnshaw, <a href="http://dx.doi.org/10.1007/s00026-002-8026-z">A constructive enumeration of meanders</a>, Ann. Comb. 6 (2002), no. 1, 7-17. [Table 1 gives first 14 rows]
%e A060972 Triangle begins:
%e A060972 1;
%e A060972 1, 1;
%e A060972 1, 6, 1;
%e A060972 1, 20, 20, 1;
%e A060972 1, 50, 160, 50, 1;
%e A060972 1, 105, 808, 808, 105, 1; ...
%Y A060972 Row sums give A005315, diagonals give A002415.
%K A060972 nonn,tabl
%O A060972 1,5
%A A060972 _F. Chapoton_, May 09 2001; extended to 14 rows, Jul 31 2011