A109158 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 last peak equal to k.

1, 1, 2, 4, 3, 1, 10, 20, 18, 12, 5, 1, 66, 132, 122, 92, 54, 24, 7, 1, 498, 996, 930, 732, 478, 264, 118, 40, 9, 1, 4066, 8132, 7634, 6140, 4214, 2552, 1342, 600, 218, 60, 11, 1, 34970, 69940, 65874, 53676, 37910, 24136, 13782, 7016, 3122, 1180, 362, 84, 13, 1

Offset: 1

Author

Emeric Deutsch, Jun 21 2005

Keywords

• nonn
• tabf

Comments

Row n has 2n terms. Row sums yield A027307. T(n,1)=A027307(n-1). T(n,2)=2*A027307(n-1) for n>=2.

Examples

			T(2,3)=3 because we have uUddd, UdUddd and Uuddd.
Triangle begins:
1,1;
2,4,3,1;
10,20,18,12,5,1;
66,132,122,92,54,24,7,1;
		

Links

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

Crossrefs

Cf. A027307.

Programs

Maple

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=t*z*(1+t)/(1-t*z-t^2*z-(1+t)*z*A-z*A^2): Gser:=simplify(series(G,z=0,10)): for n from 1 to 8 do P[n]:=coeff(Gser,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

Formula

G.f.=tz(1+t)/[1-tz-t^2z-(1+t)zA-zA^2], 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).