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A243827 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.

%I A243827 #16 Jan 24 2019 17:13:42
%S A243827 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 A243827 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 A243827 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 A243827 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 A243827 Alois P. Heinz, <a href="/A243827/b243827.txt">antidiagonals n = 0..140, flattened</a>
%e A243827 Square array A(n,k) begins:
%e A243827   0, 0, 0,  0,   0,    0,   0,    0,    0,    0, ...
%e A243827   1, 1, 1,  0,   0,    0,   0,    0,    0,    0, ...
%e A243827   0, 0, 1,  1,   1,    1,   1,    0,    0,    0, ...
%e A243827   0, 0, 1,  3,   4,    2,   4,    1,    1,    1, ...
%e A243827   0, 0, 1,  6,  11,    6,  11,    4,    5,    5, ...
%e A243827   0, 0, 1, 10,  26,   16,  26,   15,   21,   17, ...
%e A243827   0, 0, 1, 15,  57,   45,  57,   50,   78,   54, ...
%e A243827   0, 0, 1, 21, 120,  126, 120,  161,  274,  177, ...
%e A243827   0, 0, 1, 28, 247,  357, 247,  504,  927,  594, ...
%e A243827   0, 0, 1, 36, 502, 1016, 502, 1554, 3061, 1997, ...
%Y A243827 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 A243827 Main diagonal gives A243770 or column k=1 of A243752.
%Y A243827 Cf. A243753, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
%K A243827 nonn,tabl
%O A243827 0,25
%A A243827 _Alois P. Heinz_, Jun 11 2014