cp's OEIS Frontend

A304361 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive pattern UDUD, where U=(1,1) and D=(1,-1).

%I A304361 #19 Mar 25 2020 06:51:06
%S A304361 1,1,1,9,41,244,1555,10037,68599,476981,3399518,24652718,181411439,
%T A304361 1352123760,10185964435,77458698781,593871350009,4586247704944,
%U A304361 35646681303447,278665636846853,2189789189667782,17288684906561300,137081212514315262,1091163063187762414
%N A304361 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive pattern UDUD, where U=(1,1) and D=(1,-1).
%H A304361 Alois P. Heinz, <a href="/A304361/b304361.txt">Table of n, a(n) for n = 0..1072</a>
%H A304361 Vaclav Kotesovec, <a href="/A304361/a304361.txt">Recurrence (of order 6)</a>
%H A304361 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A304361 a(n) = A094507(2n,n).
%F A304361 a(n) ~ c * d^n / n^2, where d = 8.678461743575504549836851346229164298625429506253061911480810294... is the real root of equation 28*d^5 - 72*d^4 - 1119*d^3 - 3136*d^2 - 272*d - 16 = 0 and c = 0.15899091419445210968174633623072264522489566046427010886172717963... - _Vaclav Kotesovec_, Mar 25 2020
%p A304361 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
%p A304361       `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])
%p A304361         +b(x-1, y-1, [1, 3, 1, 3][t])*`if`(t=4, z, 1))))
%p A304361     end:
%p A304361 a:= n-> coeff(b(4*n, 0, 1), z, n):
%p A304361 seq(a(n), n=0..35);
%t A304361 b[x_, y_, t_] := b[x, y, t] = If[y < 0 || y > x, 0, If[x == 0, 1, Expand[ b[x - 1, y + 1, {2, 2, 4, 2}[[t]]] + b[x - 1, y - 1, {1, 3, 1, 3}[[t]]]*If[t == 4, z, 1]]]];
%t A304361 a[n_] := Coefficient[b[4*n, 0, 1], z, n];
%t A304361 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A304361 Cf. A094507, A333156.
%K A304361 nonn
%O A304361 0,4
%A A304361 _Alois P. Heinz_, May 11 2018