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 A182905 Number of weighted lattice paths in F[n]. The members of F[n] are paths of weight n that start at (0,0), do not go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. 2
 1, 1, 3, 6, 14, 32, 75, 177, 422, 1013, 2447, 5942, 14495, 35501, 87257, 215144, 531970, 1318726, 3276644, 8158736, 20354413, 50870857, 127348839, 319288920, 801657469, 2015431885, 5073224661, 12785062080, 32254748838, 81457050078 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The paths in F[n] need not end on the horizontal axis. If f(z) is the generating function of the paths in F[n] (according to weight), and g(z) is the generating function of the those paths in F[n] that end on the horizontal axis, then f = g +z^2*gf. Also, g = (1 + zg)(1+z^2*g). Eliminating g from the above two equations, one obtains f(z). LINKS M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. FORMULA G.f.: f(z)=2/[1-z-3z^2+sqrt((1+z+z^2)(1-3z+z^2))]. a(n) ~ sqrt(4935 + 2207*sqrt(5))* ((3 + sqrt(5))/2)^n / (sqrt(8*Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2016 a(n) = Sum_{k=0..n} (k+1)*Sum_{i=0..n-k+1} C(i+1,n+1-i)*C(i+1,-i+n-k))/(i+1). - Vladimir Kruchinin, Jan 25 2019 EXAMPLE a(3)=6. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hhh, hH, Hh, UD, hU, and Uh. MAPLE f := 2/(1-z-3*z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): fser := series(f, z = 0, 32): seq(coeff(fser, z, n), n = 0 .. 29); MATHEMATICA CoefficientList[Series[2/(1-x-3*x^2+Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *) PROG (Maxima) a(n):=sum((k+1)*sum((binomial(i+1, n+1-i)*binomial(i+1, -i+n-k))/(i+1), i, 0, n-k+1), k, 0, n); /* Vladimir Kruchinin, Jan 25 2019 */ CROSSREFS Sequence in context: A129954 A238768 A272362 * A330053 A192678 A114945 Adjacent sequences:  A182902 A182903 A182904 * A182906 A182907 A182908 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 16 2010 STATUS approved

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Last modified August 4 15:50 EDT 2020. Contains 336202 sequences. (Running on oeis4.)