The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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: A354841 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 27 02:41 EDT 2024. Contains 372847 sequences. (Running on oeis4.)