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%I #35 Jun 12 2014 02:11:10
%S 1,0,1,0,1,1,1,3,1,1,11,2,9,16,12,4,1,1,57,69,5,127,161,98,35,7,1,323,
%T 927,180,1515,1997,1056,280,14,4191,5539,3967,1991,781,244,64,17,1,1,
%U 10455,25638,18357,4115,220,1,20705,68850,77685,34840,5685,246,1
%N Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.
%H Alois P. Heinz, <a href="/A243752/b243752.txt">Rows n = 0..270, flattened</a>
%e Triangle T(n,k) begins:
%e : n\k : 0 1 2 3 4 5 ...
%e +-----+----------------------------------------------------------
%e : 0 : 1; [row 0 of A131427]
%e : 1 : 0, 1; [row 1 of A131427]
%e : 2 : 0, 1, 1; [row 2 of A090181]
%e : 3 : 1, 3, 1; [row 3 of A001263]
%e : 4 : 1, 11, 2; [row 4 of A091156]
%e : 5 : 9, 16, 12, 4, 1; [row 5 of A091869]
%e : 6 : 1, 57, 69, 5; [row 6 of A091156]
%e : 7 : 127, 161, 98, 35, 7, 1; [row 7 of A092107]
%e : 8 : 323, 927, 180; [row 8 of A091958]
%e : 9 : 1515, 1997, 1056, 280, 14; [row 9 of A135306]
%e : 10 : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]
%Y Columns k=0-10 give: A243754, A243770, A243771, A243772, A243773, A243774, A243775, A243776, A243777, A243778, A243779, or main diagonals of A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
%Y Row sums give A000108.
%Y Cf. A098978, A114463, A114848, A116424, A135305, A242450, A243366, A243838.
%K nonn,tabf,look
%O 0,8
%A _Alois P. Heinz_, Jun 09 2014
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