A108434 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no hills of the form ud (a hill is either a ud or a Udd starting at the x-axis).
1, 1, 7, 47, 361, 2977, 25775, 231103, 2127409, 19990241, 190957559, 1848911279, 18104425561, 178975914433, 1783843502047, 17906040994559, 180858717257185, 1836792828317761, 18745545101801063, 192145823547338927
Offset: 0
Keywords
Examples
a(2)=7 because we have uudd, uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..200
- Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
Programs
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Maple
g:=1/(1+z-z*A-z*A^2): A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3:gser:=series(g,z=0,27): 1,seq(coeff(gser,z^n),n=1..24);
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PARI
{a(n)=local(y=1+x); for(i=1, n, y=-(-1 + 3*x*y - 3*x*(1+x)*y^2 + x*(-1+2*x+x^2)*y^3) + (O(x^n))^3); polcoeff(y, n)} for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014
Formula
G.f. = 1/(1+z-zA-zA^2), where A=1+zA^2+zA^3 or, equivalently, A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
G.f. y(x) satisfies: -1+y + 3*x*y - 3*x*(1+x)*y^2 + x*(-1+2*x+x^2)*y^3 = 0. - Vaclav Kotesovec, Mar 17 2014
a(n) ~ (11+5*sqrt(5))^n * sqrt(123 + 55*sqrt(5)) / (9 * 5^(1/4) * sqrt(Pi) * n^(3/2) * 2^(n+3/2)). - Vaclav Kotesovec, Mar 17 2014
a(n) ~ phi^(5*n + 5) / (18 * 5^(1/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017
D-finite with recurrence n*(2*n+1)*(n-2)*a(n) +2*(-13*n^3+36*n^2-29*n+9)*a(n-1) +4*(n-1)*(10*n^2-20*n+9)*a(n-2) +2*(13*n^3-42*n^2+41*n-9)*a(n-3) +n*(n-2)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
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