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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

Table of n, a(n) for n=0..29.

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.)