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A108433 Triangle read by rows: T(n,k) is 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 k hills of the form ud (a hill is either a ud or a Udd starting at the x-axis). 4
1, 1, 1, 7, 2, 1, 47, 15, 3, 1, 361, 108, 24, 4, 1, 2977, 865, 184, 34, 5, 1, 25775, 7334, 1533, 276, 45, 6, 1, 231103, 64767, 13359, 2387, 385, 57, 7, 1, 2127409, 589368, 120376, 21368, 3450, 512, 70, 8, 1, 19990241, 5488033, 1112424, 196484, 31706 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums yield A027307. T(n,0)=A108434(n). A027307, A108432, A108433, A108434.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

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

FORMULA

G.f.: 1/(1-tz+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).

EXAMPLE

Example T(2,1)=2 because we have udUdd and Uddud.

Triangle begins:

1;

1,1;

7,2,1;

47,15,3,1;

361,108,24,4,1;

MAPLE

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-z*A+z-t*z-z*A^2): Gserz:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 0 to 9 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form

# second Maple program:

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

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

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

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, false)):

seq(T(n), n=0..10);  # Alois P. Heinz, Oct 06 2015

MATHEMATICA

b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, t]*If[t && y == 1, z, 1] + b[x - 1, y + 2, False] + b[x - 2, y + 1, y == 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][b[3*n, 0, False]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A027307, A108431, A108432, A108434.

Sequence in context: A060991 A120455 A268895 * A274570 A176704 A289917

Adjacent sequences:  A108430 A108431 A108432 * A108434 A108435 A108436

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 03 2005

STATUS

approved

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Last modified October 16 06:08 EDT 2019. Contains 328046 sequences. (Running on oeis4.)