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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.
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%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