A287548 - OEIS

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.

1, 2, 1, 5, 3, 2, 14, 9, 7, 4, 42, 28, 23, 16, 10, 132, 90, 76, 57, 42, 24, 429, 297, 255, 199, 156, 108, 66, 1430, 1001, 869, 695, 563, 420, 304, 174, 4862, 3432, 3003, 2442, 2019, 1568, 1210, 836, 504

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..45.

FORMULA

T(n,1) = Catalan Numbers C(n)= A000108(n).

Conjectured:

T(n,2) = C(n) - C(n-1) = A000245(n-1).

T(n,3) = C(n) - C(n-1) - C(n-2) = A067324(n-3).

T(n,4) = C(n) - C(n-1) - 2*C(n-2) - C(n-3).

T(n,n) = semi-meander solutions = A000682(n-1).

EXAMPLE

Triangle begins:

n\k 1 2 3 4 5 6 7 8

1: 1

2: 2 1

3: 5 3 2

4: 14 9 7 4

5: 42 28 23 16 10

6: 132 90 76 57 42 24

7: 429 297 255 199 156 108 66

8: 1430 1001 869 695 563 420 304 174

...

Capital letters (U,D) represent beginning and end of first and last arch. Only 1 UD ends arch sequence in next generation.

Reduction of arches: Elimination of arches:

(middle D U = new arch U D in the next arch generation)

/\ //\\ /\/\/\/\ = UDududUD

//\\/\///\\\ = UudDudUuuddD /\

/\ /\ / \

/\//\\//\\ = UDuuddUudD //\/\\ = UududD

end

For n=3 C(n)=5 nonintersecting arch configurations:

UuuddD UududD UudDUD UDUudD UDudUD T(3,1)=5

end end UDUD UDUD UudD T(3,2)=3

UD UD end T(3,3)=2

CROSSREFS

Cf. A000108, A000245, A067324, A000682.

Sequence in context: A332529 A213849 A067418 * A067323 A106534 A123346

Adjacent sequences: A287545 A287546 A287547 * A287549 A287550 A287551

KEYWORD

nonn,tabl,more

AUTHOR

Roger Ford, May 26 2017

STATUS

approved