%I #34 Aug 31 2022 02:43:42
%S 1,2,1,5,3,2,14,9,7,4,42,28,23,16,10,132,90,76,57,42,24,429,297,255,
%T 199,156,108,66,1430,1001,869,695,563,420,304,174,4862,3432,3003,2442,
%U 2019,1568,1210,836,504
%N Triangle read by rows: T(n,k), where each row begins with the Catalan number for n nonintersecting arches and transitions through k generations of eliminating and reducing arch configurations to an end row entry equal to number of semi-meander solutions for n arches.
%F T(n,1) = Catalan Numbers C(n)= A000108(n).
%F Conjectured:
%F T(n,2) = C(n) - C(n-1) = A000245(n-1).
%F T(n,3) = C(n) - C(n-1) - C(n-2) = A067324(n-3).
%F T(n,4) = C(n) - C(n-1) - 2*C(n-2) - C(n-3).
%F T(n,n) = semi-meander solutions = A000682(n-1).
%e Triangle begins:
%e n\k 1 2 3 4 5 6 7 8
%e 1: 1
%e 2: 2 1
%e 3: 5 3 2
%e 4: 14 9 7 4
%e 5: 42 28 23 16 10
%e 6: 132 90 76 57 42 24
%e 7: 429 297 255 199 156 108 66
%e 8: 1430 1001 869 695 563 420 304 174
%e ...
%e Capital letters (U,D) represent beginning and end of first and last arch. Only 1 UD ends arch sequence in next generation.
%e Reduction of arches: Elimination of arches:
%e (middle D U = new arch U D in the next arch generation)
%e /\
%e /\ //\\ /\/\/\/\ = UDududUD
%e //\\/\///\\\ = UudDudUuuddD /\
%e /\ /\ / \
%e /\//\\//\\ = UDuuddUudD //\/\\ = UududD
%e end
%e For n=3 C(n)=5 nonintersecting arch configurations:
%e UuuddD UududD UudDUD UDUudD UDudUD T(3,1)=5
%e end end UDUD UDUD UudD T(3,2)=3
%e UD UD end T(3,3)=2
%Y Cf. A000108, A000245, A067324, A000682.
%K nonn,tabl,more
%O 1,2
%A _Roger Ford_, May 26 2017