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%I #18 Jul 26 2022 12:23:55
%S 1,1,7,47,361,2977,25775,231103,2127409,19990241,190957559,1848911279,
%T 18104425561,178975914433,1783843502047,17906040994559,
%U 180858717257185,1836792828317761,18745545101801063,192145823547338927
%N 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).
%C Column 0 of A108433.
%C The radius of convergence of g.f. y(x) is r = (5*sqrt(5)-11)/2, with y(r) = (2+sqrt(5))/3. - _Vaclav Kotesovec_, Mar 17 2014
%H Vaclav Kotesovec, <a href="/A108434/b108434.txt">Table of n, a(n) for n = 0..200</a>
%H Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658: Another Type of Lattice Path</a>, American Math. Monthly, 107, 2000, 368-370.
%F 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).
%F 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
%F 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
%F 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
%F 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
%e a(2)=7 because we have uudd, uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd.
%p 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);
%o (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)}
%o for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Mar 17 2014
%Y Cf. A027307, A108431, A108432, A108433.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 03 2005
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