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A287548 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.

%I A287548 #34 Aug 31 2022 02:43:42
%S A287548 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 A287548 199,156,108,66,1430,1001,869,695,563,420,304,174,4862,3432,3003,2442,
%U A287548 2019,1568,1210,836,504
%N A287548 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 A287548 T(n,1) = Catalan Numbers C(n)= A000108(n).
%F A287548 Conjectured:
%F A287548 T(n,2) = C(n) - C(n-1) = A000245(n-1).
%F A287548 T(n,3) = C(n) - C(n-1) - C(n-2) = A067324(n-3).
%F A287548 T(n,4) = C(n) - C(n-1) - 2*C(n-2) - C(n-3).
%F A287548 T(n,n) = semi-meander solutions = A000682(n-1).
%e A287548 Triangle begins:
%e A287548 n\k    1    2    3    4    5    6    7    8
%e A287548 1:     1
%e A287548 2:     2    1
%e A287548 3:     5    3    2
%e A287548 4:     14   9    7    4
%e A287548 5:     42   28   23   16   10
%e A287548 6:     132  90   76   57   42   24
%e A287548 7:     429  297  255  199  156  108  66
%e A287548 8:     1430 1001 869  695  563  420  304  174
%e A287548 ...
%e A287548 Capital letters (U,D) represent beginning and end of first and last arch. Only 1 UD ends arch sequence in next generation.
%e A287548 Reduction of arches:            Elimination of arches:
%e A287548 (middle D U = new arch U D in the next arch generation)
%e A287548             /\
%e A287548      /\    //\\                      /\/\/\/\  = UDududUD
%e A287548     //\\/\///\\\  = UudDudUuuddD        /\
%e A287548         /\  /\                         /  \
%e A287548      /\//\\//\\   =  UDuuddUudD       //\/\\   =  UududD
%e A287548                                         end
%e A287548 For n=3 C(n)=5  nonintersecting arch configurations:
%e A287548    UuuddD   UududD   UudDUD   UDUudD   UDudUD   T(3,1)=5
%e A287548     end      end      UDUD     UDUD     UudD    T(3,2)=3
%e A287548                        UD       UD       end    T(3,3)=2
%Y A287548 Cf. A000108, A000245, A067324, A000682.
%K A287548 nonn,tabl,more
%O A287548 1,2
%A A287548 _Roger Ford_, May 26 2017