A243965 Number of Dyck paths of semilength n such that both consecutive patterns of Dyck paths of semilength 2 occur at least once.

0, 0, 0, 0, 2, 10, 44, 179, 702, 2701, 10278, 38866, 146450, 550817, 2070116, 7779655, 29248932, 110047905, 414446256, 1562538171, 5898049688, 22290789562, 84351810044, 319609669957, 1212552963576, 4606078246284, 17518748817596, 66712192842068, 254346235738120

Offset: 0

Author

Alois P. Heinz, Jun 16 2014

Keywords

• nonn

Comments

The consecutive patterns 1010, 1100 are counted. Here 1=Up=(1,1), 0=Down=(1,-1).

Examples

			a(4) = 2: 10101100, 11001010.
a(5) = 10: 1010101100, 1010110010, 1010111000, 1011001010, 1100101010, 1100110100, 1101001100, 1101011000, 1110001010, 1110010100.
Here 1=Up=(1,1), 0=Down=(1,-1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Recurrence (of order 10)

Crossrefs

Cf. A014486, A063171, A243820, A243966.

Programs

Maple

b:= proc(x, y, t, s) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, `if`(s={}, 1, 0), `if`(nops(s)>x, 0, add(
      b(x-1, y-1+2*j, irem(2*t+j, 8), s minus {2*t+j}), j=0..1))))
    end:
a:= n->  b(2*n, 0, 0, {10, 12}):
seq(a(n), n=0..30);

Mathematica

b[x_, y_, t_, s_] := b[x, y, t, s] = If[y<0 || y>x, 0, If[x == 0, If[s == {}, 1, 0], If[Length[s] > x, 0, Sum[b[x - 1, y - 1 + 2 j, Mod[2t + j, 8], s ~Complement~ {2t + j}], {j, 0, 1}]]]];
a[n_] := b[2n, 0, 0, {10, 12}];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

a(n) ~ 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 18 2014