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

0,5

Comments

The consecutive patterns 1010, 1100 are counted. 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)

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A014486, A063171, A243820, A243966.

Sequence in context: A025590 A122932 A080069 * A218780 A068551 A099919

Adjacent sequences: A243962 A243963 A243964 * A243966 A243967 A243968

Keyword

nonn

Author

Alois P. Heinz, Jun 16 2014

Status

approved