

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
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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



FORMULA

G.f.: f(z)=2/[1z3z^2+sqrt((1+z+z^2)(13z+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. Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 8) / (2*sqrt(Pi)*n^(3/2)), where phi = A001622 is the golden ratio.  Vaclav Kotesovec, Dec 06 2021
a(n) = Sum_{k=0..n} (k+1)*Sum_{i=0..nk+1} C(i+1,n+1i)*C(i+1,i+nk))/(i+1).  Vladimir Kruchinin, Jan 25 2019
Dfinite with recurrence (n+2)*a(n) +(4*n5)*a(n1) +(n1)*a(n2) +(4*n+5)*a(n3) +(7*n16)*a(n4) +2*(n1)*a(n5) +2*(n+4)*a(n6)=0.  R. J. Mathar, Jul 24 2022


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/(1z3*z^2+sqrt((1+z+z^2)*(13*z+z^2))): fser := series(f, z = 0, 32): seq(coeff(fser, z, n), n = 0 .. 29);


MATHEMATICA

CoefficientList[Series[2/(1x3*x^2+Sqrt[(1+x+x^2)*(13*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)


PROG

(Maxima)
a(n):=sum((k+1)*sum((binomial(i+1, n+1i)*binomial(i+1, i+nk))/(i+1), i, 0, nk+1), k, 0, n); /* Vladimir Kruchinin, Jan 25 2019 */


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



