%I #40 Jun 06 2023 18:09:27
%S 1,1,2,4,11,31,92,283,893,2875,9407,31189,104555,353794,1206821,
%T 4145350,14326184,49778473,173794610,609392578,2145057797,7577098816,
%U 26850456704,95425761829,340047930692,1214738997142,4349231444405,15604726428805,56098211626478
%N Number of Dyck paths of semilength n avoiding the consecutive step pattern UDUDU, where U=(1,1) and D=(1,-1).
%H Alois P. Heinz, <a href="/A247333/b247333.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H Beáta Bényi, Toufik Mansour, and José L. Ramírez, <a href="https://ajc.maths.uq.edu.au/pdf/84/ajc_v84_p325.pdf">Set partitions and non-crossing partitions with l-neighbors and l-isolated elements</a>, Australasian J. Comb. (2022) Vol. 84, No. 2, 325-340.
%F Recursion: see Maple program.
%F a(n) ~ sqrt(42-6*sqrt(21)) * ((3+sqrt(21))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 16 2014
%F From _Vladimir Kruchinin_, Mar 03 2016: (Start)
%F G.f.: (1+x+x^2-sqrt(-3*x^4-6*x^3-5*x^2-2*x+1))/(2*x^2+2*x).
%F G.f.: B(x)+1, where B(x) satisfies B(x)=(x+x^2)*(1+B(x)+B(x)^2).
%F a(n) = Sum_{k=1..n} binomial(k,n-k)*M(k+1), a(0)=1, where M(k) are Motzkin numbers (A001006). (End)
%p a:= proc(n) option remember; `if`(n<5, [1$2, 2, 4, 11][n+1],
%p ((n-2)*a(n-1) +(7*n-11)*a(n-2) +(11*n-31)*a(n-3)
%p +(9*n-36)*a(n-4) +(3*n-15)*a(n-5)) / (n+1))
%p end:
%p seq(a(n), n=0..40);
%t CoefficientList[Series[(1 + x + x^2 - Sqrt[-3 x^4 - 6 x^3 - 5 x^2 - 2 x + 1])/(2 x^2 + 2 x), {x, 0, 28}], x] (* or *)
%t {1}~Join~Table[Sum[(Binomial[k, n - k] Sum[Binomial[j, -k + 2 j - 1] Binomial[k, j], {j, 0, k}])/k, {k, 1, n}], {n, 1, 28}] (* _Michael De Vlieger_, Mar 03 2016, the latter after Maxima by _Vladimir Kruchinin_ *)
%o (Maxima)
%o a(n):=if n=0 then 1 else sum((binomial(k,n-k)*sum(binomial(j,-k+2*j-1)*binomial(k,j),j,0,k))/k,k,1,n); /* _Vladimir Kruchinin_, Mar 03 2016 */
%Y Cf. A001006.
%Y Column k=0 of A246188.
%Y Column k=21 of A243753.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 13 2014