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A243753 Number A(n,k) of Dyck paths of semilength n avoiding 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 A243753 #28 Jan 24 2019 17:11:29
%S A243753 1,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,1,1,0,
%T A243753 0,0,1,1,1,2,1,1,0,0,0,1,1,2,1,4,1,1,0,0,0,1,1,2,4,1,9,1,1,0,0,0,1,1,
%U A243753 2,4,9,1,21,1,1,0,0,0,1,1,1,4,9,21,1,51,1,1,0,0,0
%N A243753 Number A(n,k) of Dyck paths of semilength n avoiding 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 A243753 Alois P. Heinz, <a href="/A243753/b243753.txt">Antidiagonals n = 0..140, flattened</a>
%e A243753 Square array A(n,k) begins:
%e A243753   1, 1, 1, 1, 1,   1, 1,   1,   1,    1, ...
%e A243753   0, 0, 0, 1, 1,   1, 1,   1,   1,    1, ...
%e A243753   0, 0, 0, 1, 1,   1, 1,   2,   2,    2, ...
%e A243753   0, 0, 0, 1, 1,   2, 1,   4,   4,    4, ...
%e A243753   0, 0, 0, 1, 1,   4, 1,   9,   9,    9, ...
%e A243753   0, 0, 0, 1, 1,   9, 1,  21,  21,   23, ...
%e A243753   0, 0, 0, 1, 1,  21, 1,  51,  51,   63, ...
%e A243753   0, 0, 0, 1, 1,  51, 1, 127, 127,  178, ...
%e A243753   0, 0, 0, 1, 1, 127, 1, 323, 323,  514, ...
%e A243753   0, 0, 0, 1, 1, 323, 1, 835, 835, 1515, ...
%p A243753 A:= proc(n, k) option remember; local b, m, r, h;
%p A243753       if k<2 then return `if`(n=0, 1, 0) fi;
%p A243753       m:= iquo(k, 2, 'r'); h:= 2^ilog2(k); b:=
%p A243753       proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
%p A243753         `if`(t=m and r=1, 0, b(x-1, y+1, irem(2*t+1, h)))+
%p A243753         `if`(t=m and r=0, 0, b(x-1, y-1, irem(2*t, h)))))
%p A243753       end; forget(b);
%p A243753       b(2*n, 0, 0)
%p A243753     end:
%p A243753 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A243753 A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k<2, Return[If[n == 0, 1, 0]]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, If[t == m && r == 1, 0, b[x-1, y+1, Mod[2*t+1, h]]] + If[t == m && r == 0, 0, b[x-1, y-1, Mod[2*t, h]]]]]; b[2*n, 0, 0]]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 27 2015, after _Alois P. Heinz_ *)
%Y A243753 Columns give: 0, 1, 2: A000007, 3, 4, 6: A000012, 5: A001006(n-1) for n>0, 7, 8, 14: A001006, 9: A135307, 10: A078481 for n>0, 11, 13: A105633(n-1) for n>0, 12: A082582, 15, 16: A036765, 19, 27: A114465, 20, 24, 26: A157003, 21: A247333, 25: A187256(n-1) for n>0.
%Y A243753 Main diagonal gives A243754 or column k=0 of A243752.
%Y A243753 Cf. A242450, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
%K A243753 nonn,tabl
%O A243753 0,40
%A A243753 _Alois P. Heinz_, Jun 09 2014