OFFSET
0,2
COMMENTS
Also number of Grand Dyck paths of length 2*n with no zigzags, that is, with no factors UDU or DUD. [Emanuele Munarini, Jul 07 2011]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..220
Andrei Asinowski, Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
T. Doslic, Seven lattice paths to log-convexity, Acta Appl. Mathem. 110 (3) (2010) 1373-139, eq 4.
E. Munarini and N. Z. Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, arXiv:math/0512548 [math.CO], 2007.
FORMULA
G.f.: sqrt( ( 1 + x + x^2 ) / ( 1 - 3*x + x^2 ) ).
a(n) = sum( binomial( n - k + 2*floor(k/3), floor(k/3) )^2, k=0..n+floor(n/2)).
a(n) = sum( binomial(n-k,k)^2*( 2*n^2 - 6*n*k + 6*k^2 )/(n-k)^2 ), k=0..floor(n/2) ).
a(n) ~ 2 * ((3+sqrt(5))/2)^n / (5^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
a(n) = [x^n y^n](1+x*y+x^2*y^2)/(1-x-y+x*y-x^2*y^2). - Gheorghe Coserea, Jul 18 2016
D-finite with recurrence: n*a(n) -2*n*a(n-1) +(-n+2)*a(n-2) +2*(-n+4)*a(n-3) +(n-4)*a(n-4)=0. [Doslic] - R. J. Mathar, Jun 21 2018
EXAMPLE
For n = 2 : 0011, 0110, 1001, 1100.
For n = 3 : 000111, 011001, 100011, 110001, 001110, 011100, 100110, 111000.
MAPLE
a:= proc(n) option remember; `if`(n<5, [1, 2, 4, 8, 18][n+1],
(2*n*a(n-1)+(n-2)*a(n-2)+(2*n-8)*a(n-3)-(n-4)*a(n-4))/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Feb 13 2020
MATHEMATICA
Table[SeriesCoefficient[Series[Sqrt[(1 + x + x^2)/(1 - 3 x + x^2)], {x, 0, n}], n], {n, 0, 40}]
PROG
(Maxima) a(n):=coeff(taylor((1+x+x^2)/sqrt(1-2*x-x^2-2*x^3+x^4), x, 0, n), x, n);
makelist(a(n), n, 0, 12); [Emanuele Munarini, Jul 07 2011]
(PARI) x='x+O('x^99); Vec(((1+x+x^2)/(1-3*x+x^2))^(1/2)) \\ Altug Alkan, Jul 18 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Dec 17 2002
STATUS
approved