A108437 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 height of the first peak equal to k.

1, 1, 2, 5, 2, 1, 10, 28, 13, 11, 3, 1, 66, 196, 90, 89, 34, 18, 4, 1, 498, 1532, 694, 736, 311, 197, 66, 26, 5, 1, 4066, 12804, 5738, 6344, 2800, 1937, 762, 367, 110, 35, 6, 1, 34970, 111964, 49758, 56576, 25560, 18636, 7953, 4263, 1551, 615, 167, 45, 7, 1

Offset

1,3

Comments

Row n contains 2n terms. Row sums yield A027307. T(n,1)=A027307(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-tzA-t^2*zA^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;
2,5,2,1;
10,28,13,11,3,1;
66,196,90,89,34,18,4,1;

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

Crossrefs

Cf. A027307.

Sequence in context: A180957 A124780 A369872 * A226029 A152765 A327867

Adjacent sequences: A108434 A108435 A108436 * A108438 A108439 A108440

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 04 2005

Status

approved