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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. 24
 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, 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, 2, 4, 9, 1, 21, 1, 1, 0, 0, 0, 1, 1, 1, 4, 9, 21, 1, 51, 1, 1, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,40 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened EXAMPLE Square array A(n,k) begins:   1, 1, 1, 1, 1,   1, 1,   1,   1,    1, ...   0, 0, 0, 1, 1,   1, 1,   1,   1,    1, ...   0, 0, 0, 1, 1,   1, 1,   2,   2,    2, ...   0, 0, 0, 1, 1,   2, 1,   4,   4,    4, ...   0, 0, 0, 1, 1,   4, 1,   9,   9,    9, ...   0, 0, 0, 1, 1,   9, 1,  21,  21,   23, ...   0, 0, 0, 1, 1,  21, 1,  51,  51,   63, ...   0, 0, 0, 1, 1,  51, 1, 127, 127,  178, ...   0, 0, 0, 1, 1, 127, 1, 323, 323,  514, ...   0, 0, 0, 1, 1, 323, 1, 835, 835, 1515, ... MAPLE A:= proc(n, k) option remember; local b, m, r, h;       if k<2 then return `if`(n=0, 1, 0) fi;       m:= iquo(k, 2, 'r'); h:= 2^ilog2(k); b:=       proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,         `if`(t=m and r=1, 0, b(x-1, y+1, irem(2*t+1, h)))+         `if`(t=m and r=0, 0, b(x-1, y-1, irem(2*t, h)))))       end; forget(b);       b(2*n, 0, 0)     end: seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA 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 *) CROSSREFS 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. Main diagonal gives A243754 or column k=0 of A243752. Cf. A242450, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836. Sequence in context: A284256 A339772 A250211 * A219238 A025918 A030425 Adjacent sequences:  A243750 A243751 A243752 * A243754 A243755 A243756 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 09 2014 STATUS approved

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Last modified April 14 03:49 EDT 2021. Contains 342941 sequences. (Running on oeis4.)