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 A078678 Number of binary strings with n 1's and n 0's avoiding zigzags, that is avoiding the substrings 101 and 010. 6
 1, 2, 4, 8, 18, 42, 100, 242, 592, 1460, 3624, 9042, 22656, 56970, 143688, 363348, 920886, 2338566, 5949148, 15157874, 38674978, 98803052, 252701484, 646990518, 1658066668, 4252908542, 10917422860, 28046438252, 72099983802, 185469011130, 477383400300 (list; graph; refs; listen; history; text; internal format)
 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, SÃ©minaire 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 Cf. A078679, A128588. Cf. A003440. Main diagonal of array A099172. Related to diagonal of rational functions: A268545-A268555. Sequence in context: A057151 A026699 A182780 * A261492 A027056 A024428 Adjacent sequences:  A078675 A078676 A078677 * A078679 A078680 A078681 KEYWORD nonn AUTHOR Emanuele Munarini, Dec 17 2002 STATUS approved

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Last modified July 30 03:29 EDT 2021. Contains 346347 sequences. (Running on oeis4.)