%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/10.1016/0012365X(93)E0147V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155176.
%H D. E. Knuth, <a href="http://wwwcsfaculty.stanford.edu/~uno/flaj2014.pdf">Problems That Philippe Would Have Loved</a>, Paris 2014.
%F G.f. = G = G(t, z) satisfies z(1tz)G^2(1+z2tz)G+1tz = 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
