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%I #16 Jan 24 2019 17:13:42
%S 0,0,1,0,1,0,0,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,3,1,0,0,0,0,1,4,6,1,
%T 0,0,0,0,1,2,11,10,1,0,0,0,0,0,4,6,26,15,1,0,0,0,0,0,1,11,16,57,21,1,
%U 0,0,0,0,0,1,4,26,45,120,28,1,0,0,0,0,1,1,5,15,57,126,247,36,1,0,0
%N Number A(n,k) of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A243827/b243827.txt">antidiagonals n = 0..140, flattened</a>
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
%e 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, ...
%e 0, 0, 1, 3, 4, 2, 4, 1, 1, 1, ...
%e 0, 0, 1, 6, 11, 6, 11, 4, 5, 5, ...
%e 0, 0, 1, 10, 26, 16, 26, 15, 21, 17, ...
%e 0, 0, 1, 15, 57, 45, 57, 50, 78, 54, ...
%e 0, 0, 1, 21, 120, 126, 120, 161, 274, 177, ...
%e 0, 0, 1, 28, 247, 357, 247, 504, 927, 594, ...
%e 0, 0, 1, 36, 502, 1016, 502, 1554, 3061, 1997, ...
%Y Columns k=2-10 give: A000012(n) for n>0, A000217(n-1) for n>0, A000295(n-1) for n>0, A005717(n-1) for n>1, A000295(n-1) for n>0, A014532(n-2) for n>2, A108863, A244235, A244236.
%Y Main diagonal gives A243770 or column k=1 of A243752.
%Y Cf. A243753, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
%K nonn,tabl
%O 0,25
%A _Alois P. Heinz_, Jun 11 2014
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