A243966 Number of Dyck paths of semilength n such that all five consecutive patterns of Dyck paths of semilength 3 occur at least once.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 138, 1152, 8166, 52098, 308964, 1733444, 9311300, 48280464, 243112106, 1194286106, 5744306228, 27129749648, 126111332862, 578106334718, 2617667137358, 11723920607410, 51998857149406, 228621028644376, 997286152915772

Offset

0,13

Comments

The five consecutive patterns that occur at least once each are 101010, 101100, 110010, 110100, 111000. Here 1=Up=(1,1), 0=Down=(1,-1).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Example

a(12) = 12: 101010110010110100111000, 101010110010111000110100, 101100101010110100111000, 101100101010111000110100, 110100101010110010111000, 110100101100101010111000, 110100111000101010110010, 110100111000101100101010, 111000101010110010110100, 111000101100101010110100, 111000110100101010110010, 111000110100101100101010.
Here 1=Up=(1,1), 0=Down=(1,-1).

Maple

b:= proc(x, y, l) option remember; local m; m:= min(l[]);
      `if`(y>x or y<0 or 7-m>x, 0, `if`(x=0, 1,
      b(x-1, y+1, [[2, 3, 4, 4, 2, 2, 7][l[1]],
       [2, 3, 3, 5, 3, 2, 7][l[2]], [2, 3, 3, 2, 6, 3, 7][l[3]],
       [2, 2, 4, 5, 2, 4, 7][l[4]], [2, 2, 4, 2, 6, 2, 7][l[5]]])+
      b(x-1, y-1, [[1, 1, 1, 5, 6, 7, 7][l[1]],
       [1, 1, 4, 1, 6, 7, 7][l[2]], [1, 1, 4, 5, 1, 7, 7][l[3]],
       [1, 3, 1, 3, 6, 7, 7][l[4]], [1, 3, 1, 5, 1, 7, 7][l[5]]])))
    end:
a:= n-> b(2*n, 0, [1$5]):
seq(a(n), n=0..35);

Mathematica

b[x_, y_, l_] := b[x, y, l] = Module[{m = Min[l]},
        If[y>x || y<0 || 7-m>x, 0, If[x == 0, 1,
        b[x-1, y+1, MapIndexed[#1[[l[[#2[[1]] ]] ]]&,
        {{2, 3, 4, 4, 2, 2, 7},
         {2, 3, 3, 5, 3, 2, 7},
         {2, 3, 3, 2, 6, 3, 7},
         {2, 2, 4, 5, 2, 4, 7},
         {2, 2, 4, 2, 6, 2, 7}}]]] +
        b[x-1, y-1, MapIndexed[#1[[l[[#2[[1]] ]] ]]&,
        {{1, 1, 1, 5, 6, 7, 7},
         {1, 1, 4, 1, 6, 7, 7},
         {1, 1, 4, 5, 1, 7, 7},
         {1, 3, 1, 3, 6, 7, 7},
         {1, 3, 1, 5, 1, 7, 7}}]]]];
a[n_] := b[2n, 0, {1, 1, 1, 1, 1}];
a /@ Range[0, 35] (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)

Crossrefs

Cf. A014486, A063171, A243820, A243965, A243986.

Sequence in context: A264503 A000467 A059517 * A097167 A125469 A113366

Adjacent sequences: A243963 A243964 A243965 * A243967 A243968 A243969

Keyword

nonn

Author

Alois P. Heinz, Jun 16 2014

Status

approved