A108438 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 having abscissa of the first peak equal to k.

1, 1, 4, 3, 2, 1, 24, 18, 13, 7, 3, 1, 172, 130, 96, 55, 28, 12, 4, 1, 1360, 1034, 772, 458, 249, 119, 50, 18, 5, 1, 11444, 8738, 6568, 3982, 2244, 1137, 526, 219, 80, 25, 6, 1, 100520, 76994, 58140, 35770, 20624, 10836, 5293, 2383, 981, 365, 119, 33, 7, 1

Offset

1,3

Comments

Row n contains 2n terms. Row sums yield A027307. T(n,1)=A032349(n-1).

Links

Table of n, a(n) for n=1..56.

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

Formula

G.f.: G = G(t,z) = 1/(1-t^2zA-tzA^2)-1, where A=1+zA^2+zA^3=(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

T(2,3) = 2 because we have Uuddd and uUddd.
Triangle begins:
1,1;
4,3,2,1;
24,18,13,7,3,1;
172,130,96,55,28,12,4,1;

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-t^2*z*A-t*z*A^2)-1: Gserz:=simplify(series(G, z=0, 10)): for n from 1 to 8 do P[n]:=sort(coeff(Gserz, z^n)) od: > for n from 1 to 8 do seq(coeff(P[n], t^k), k=1..2*n) od; # yields sequence in triangular form

Crossrefs

Cf. A027307, A032349.

Sequence in context: A129154 A055115 A294280 * A347738 A082504 A237524

Adjacent sequences: A108435 A108436 A108437 * A108439 A108440 A108441

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 04 2005

Status

approved