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 A091866 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having pyramid weight k. 16

%I

%S 1,0,1,0,0,2,0,0,1,4,0,0,1,5,8,0,0,1,7,18,16,0,0,1,9,34,56,32,0,0,1,

%T 11,55,138,160,64,0,0,1,13,81,275,500,432,128,0,0,1,15,112,481,1205,

%U 1672,1120,256,0,0,1,17,148,770,2471,4797,5264,2816,512,0,0,1,19,189,1156,4536,11403,17738,15808,6912,1024

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having pyramid weight k.

%C A pyramid in a Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.

%C Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 0, 1, 0, 0, 1, 0, 0, 1, ...](periodic sequence 0,0,1) DELTA [1, 1, 0, 1, 1, 0, 1, 1, 0, ...](periodic sequence 1,1,0), where DELTA is the operator defined in A084938 . - _Philippe Deléham_, Aug 18 2006

%C _Peter Luschny_ observes that one of the rows of this triangle seems to appear on page 26 of Knuth (2014). - _N. J. A. Sloane_, Aug 02 2014

%H Alois P. Heinz, <a href="/A091866/b091866.txt">Rows n = 0..140, flattened</a>

%H A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.

%H D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~uno/flaj2014.pdf">Problems That Philippe Would Have Loved</a>, Paris 2014.

%F G.f. = G = G(t, z) satisfies z(1-tz)G^2-(1+z-2tz)G+1-tz = 0.

%F Sum_{k=0..n} T(n,k) = A000108(n). - _Philippe Deléham_, Aug 18 2006

%e T(4,3)=5 because the Dyck paths of semilength 4 having pyramid weight 3 are: (ud)u(ud)(ud)d, u(ud)(ud)d(ud), u(ud)(ud)(ud)d, u(ud)(uudd)d and u(uudd)(ud)d [here u=(1,1), d=(1,-1) and the maximal pyramids, of total length 3, are shown between parentheses].

%e Triangle begins:

%e .[1],

%e .[0, 1],

%e .[0, 0, 2],

%e .[0, 0, 1, 4],

%e .[0, 0, 1, 5, 8],

%e .[0, 0, 1, 7, 18, 16],

%e .[0, 0, 1, 9, 34, 56, 32],

%e .[0, 0, 1, 11, 55, 138, 160, 64]

%e .[0, 0, 1, 13, 81, 275, 500, 432, 128]

%e ...

%K nonn,tabl

%O 0,6

%A _Emeric Deutsch_, Mar 10 2004

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