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 A129400 Number of walks of length n on one 60-degree wedge of the equilateral triangular lattice. The walk can go along the walls of the wedge, but cannot cross the walls. 7
 1, 2, 8, 32, 144, 672, 3264, 16256, 82688, 427520, 2240512, 11874304, 63533056, 342712320, 1861779456, 10176823296, 55932813312, 308907737088, 1713473323008, 9541666209792, 53322206674944, 298943898451968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Counts colored Motzkin paths where each of the steps has two possible colors. Series reversion of x/(1+2x+4x^2). - Paul Barry, Sep 04 2007 Hankel transform is 4^C(n+1,2). - Paul Barry, Oct 01 2009 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013. Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017. Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017) Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020. FORMULA a(n) = 2^n*A001006(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k)*2^k*2^k where C(n) = A000108(n). - Paul Barry, Sep 04 2007 G.f.: 1/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-.... (continued fraction). - Paul Barry, Oct 01 2009 G.f.: (1/(8*x^2)) * (1-2*x-(1-4*x-12*x^2)^(1/2)). - Mark van Hoeij, Nov 02 2009 E.g.f.: a(n) = n! * [x^n] exp(2*x)*BesselI(1,4*x)/(2*x). - Peter Luschny, Aug 25 2012 Recurrence: (n+2)*a(n) = 2*(2*n+1)*a(n-1) + 12*(n-1)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012 a(n) ~ 3*sqrt(3)*6^n/(2*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012 a(n) = 2^n*GegenbauerC(n, -n-1, -1/2)/(n+1). - Peter Luschny, May 09 2016 G.f.: A(x) = 1/(1 + 2*x)*c(2*x/(1 + 2*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Cf. A005572. - Peter Bala, Aug 18 2021 EXAMPLE a(1) = 2 because we can go east or northeast. MAPLE countwalk2 := proc (i::integer, j::integer, n::integer) option remember: if n < 0 or j < 0 or i < j then 0 elif n = 0 and i = 0 and j = 0 then 1 elif n = 0 then 0 else procname(i-2, j, n-1)+procname(i+2, j, n-1)+procname(i-1, j+1, n-1)+procname(i+1, j+1, n-1)+procname(i+1, j-1, n-1)+procname(i-1, j-1, n-1) end if end proc: counter2 := proc (n::nonnegint) option remember: add(add(countwalk2(i, j, n), i = 0 .. 2*n), j = 0 .. n) end proc: g := n -> simplify(2^n*GegenbauerC(n, -n-1, -1/2)/(n+1)): seq(g(n), n=0..21); # Peter Luschny, May 09 2016 T := proc(n, k) option remember; if n < 0 or k < 0 then 0 elif n = 0 then binomial(2*k, k)/(k+1) else 2*(T(n-1, k+1) - T(n-1, k)) fi end: a := n -> T(n, 1): seq(a(n), n=0..21); # Peter Luschny, Aug 23 2017 MATHEMATICA CoefficientList[Series[1/(8*x^2)*(1-2*x-Sqrt[1-4*x-12*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *) CROSSREFS Cf. A000108, A005572. Sequence in context: A150862 A150863 A084137 * A003304 A150864 A202814 Adjacent sequences: A129397 A129398 A129399 * A129401 A129402 A129403 KEYWORD nonn,walk,easy AUTHOR Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 28 2007 STATUS approved

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Last modified September 16 22:04 EDT 2024. Contains 375979 sequences. (Running on oeis4.)