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

Author

Alois P. Heinz, Jun 16 2014

Keywords

• nonn

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).

Examples

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

Links

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

Crossrefs

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

Programs

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 *)