OFFSET
0,2
COMMENTS
For n >= 2, a(n-2) is the number of 021-avoiding ascent sequences of length n with exactly one occurrence of the consecutive pattern 01. For example, with n=3, a(1)=4 counts 001, 010, 011, 012. - David Callan, Nov 13 2019
REFERENCES
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From N. J. A. Sloane, May 09 2012 [An early version on the arXiv had A014043 instead of A014143]
Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv, arXiv:1302.2274 [math.CO], 2013)
FORMULA
G.f.: (1-2*z-sqrt(1-4*z))/(2*z^2*(1-z)^2). - Emeric Deutsch, Jan 27 2003
Recurrence: (n+2)*a(n) = 6*(n+1)*a(n-1) - 3*(3*n+2)*a(n-2) + 2*(2*n+1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ 2^(2n+6)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012
a(n) = 2 * Sum_{k=0..n} Sum_{j=0..k} C(2*j+1,j)/(j+2). - Vaclav Kotesovec, Oct 27 2012
MATHEMATICA
Table[SeriesCoefficient[(1-2*x-Sqrt[1-4*x])/(2*x^2*(1-x)^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[2*Sum[Sum[Binomial[2*j+1, j]/(j+2), {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 27 2012 *)
PROG
(PARI) x='x+O('x^66); Vec((1-2*x-sqrt(1-4*x))/(2*x^2*(1-x)^2)) \\ Joerg Arndt, May 04 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved