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A108432 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 (a hill is either a ud or a Udd starting at the x-axis). 4
1, 0, 6, 34, 274, 2266, 19738, 177642, 1640050, 15445690, 147813706, 1433309194, 14052298690, 139063589370, 1387288675002, 13936344557354, 140859338668306, 1431424362057018, 14616361066692778, 149892742974500042 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Column 0 of A108431.

The radius of convergence of g.f. y(x) is r = (5*sqrt(5)-11)/2, with y(r) = (11*sqrt(5)+23)/38. - Vaclav Kotesovec, Mar 17 2014

LINKS

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..960 (first 151 terms from Vaclav Kotesovec)

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

FORMULA

G.f.: 1/(1+2z-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 + 6*x*y - 5*x*y^2 - 12*x^2*y^2 - x*y^3 + 6*x^2*y^3 + 8*x^3*y^3 = 0. - Vaclav Kotesovec, Mar 17 2014

a(n) ~ (11+5*sqrt(5))^n * sqrt(273965 + 122523*sqrt(5)) / (361 * sqrt(5*Pi) * n^(3/2) * 2^(n+3/2)). - Vaclav Kotesovec, Mar 17 2014

EXAMPLE

a(2)=6 because we have uudd, uUddd, Ududd, UdUddd, Uuddd and UUdddd.

MAPLE

g:=1/(1+2*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);

# second Maple program:

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,

     `if`(x=0, 1, `if`(t and y=1, 0, b(x-1, y-1, t))+

      b(x-1, y+2, is(y=0))+b(x-2, y+1, is(y=0))))

    end:

a:= n-> b(3*n, 0, false):

seq(a(n), n=0..25);  # Alois P. Heinz, Oct 06 2015

MATHEMATICA

CoefficientList[Series[9/(3 + 18*x + 2*(3+x)*Cos[2/3*ArcSin[Sqrt[x]*(18+x)/(3+x)^(3/2)]] - 2*x*Sqrt[(3+x)/x]*Sin[1/3*ArcSin[Sqrt[x]*(18+x)/(3+x)^(3/2)]]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 17 2014 *)

PROG

(PARI) {a(n)=local(y=1); for(i=1, n, y=-(-1 + 6*x*y - 5*x*y^2 - 12*x^2*y^2 - x*y^3 + 6*x^2*y^3 + 8*x^3*y^3) + (O(x^n))^3); polcoeff(y, n)}

for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014

CROSSREFS

Cf. A027307, A108431, A108433.

Sequence in context: A197436 A302148 A218685 * A245466 A284330 A125759

Adjacent sequences:  A108429 A108430 A108431 * A108433 A108434 A108435

KEYWORD

nonn,nice

AUTHOR

Emeric Deutsch, Jun 03 2005

STATUS

approved

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)