A243965 - OEIS

%I #19 Dec 21 2020 07:15:56

%S 0,0,0,0,2,10,44,179,702,2701,10278,38866,146450,550817,2070116,

%T 7779655,29248932,110047905,414446256,1562538171,5898049688,

%U 22290789562,84351810044,319609669957,1212552963576,4606078246284,17518748817596,66712192842068,254346235738120

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

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

%H Alois P. Heinz, <a href="/A243965/b243965.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="/A243965/a243965.txt">Recurrence (of order 10)</a>

%F a(n) ~ 4^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 18 2014

%e a(4) = 2: 10101100, 11001010.

%e a(5) = 10: 1010101100, 1010110010, 1010111000, 1011001010, 1100101010, 1100110100, 1101001100, 1101011000, 1110001010, 1110010100.

%e Here 1=Up=(1,1), 0=Down=(1,-1).

%p b:= proc(x, y, t, s) option remember; `if`(y<0 or y>x, 0,

%p       `if`(x=0, `if`(s={}, 1, 0), `if`(nops(s)>x, 0, add(

%p       b(x-1, y-1+2*j, irem(2*t+j, 8), s minus {2*t+j}), j=0..1))))

%p     end:

%p a:= n->  b(2*n, 0, 0, {10, 12}):

%p seq(a(n), n=0..30);

%t 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}]]]];

%t a[n_] := b[2n, 0, 0, {10, 12}];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)

%Y Cf. A014486, A063171, A243820, A243966.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Jun 16 2014