A247333 Number of Dyck paths of semilength n avoiding the consecutive step pattern UDUDU, where U=(1,1) and D=(1,-1).
1, 1, 2, 4, 11, 31, 92, 283, 893, 2875, 9407, 31189, 104555, 353794, 1206821, 4145350, 14326184, 49778473, 173794610, 609392578, 2145057797, 7577098816, 26850456704, 95425761829, 340047930692, 1214738997142, 4349231444405, 15604726428805, 56098211626478
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
- Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
- Beáta Bényi, Toufik Mansour, and José L. Ramírez, Set partitions and non-crossing partitions with l-neighbors and l-isolated elements, Australasian J. Comb. (2022) Vol. 84, No. 2, 325-340.
Programs
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Maple
a:= proc(n) option remember; `if`(n<5, [1$2, 2, 4, 11][n+1], ((n-2)*a(n-1) +(7*n-11)*a(n-2) +(11*n-31)*a(n-3) +(9*n-36)*a(n-4) +(3*n-15)*a(n-5)) / (n+1)) end: seq(a(n), n=0..40); -
Mathematica
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 *) {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 *) -
Maxima
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 */
Formula
Recursion: see Maple program.
a(n) ~ sqrt(42-6*sqrt(21)) * ((3+sqrt(21))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 16 2014
From Vladimir Kruchinin, Mar 03 2016: (Start)
G.f.: (1+x+x^2-sqrt(-3*x^4-6*x^3-5*x^2-2*x+1))/(2*x^2+2*x).
G.f.: B(x)+1, where B(x) satisfies B(x)=(x+x^2)*(1+B(x)+B(x)^2).
a(n) = Sum_{k=1..n} binomial(k,n-k)*M(k+1), a(0)=1, where M(k) are Motzkin numbers (A001006). (End)