A108451 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 pyramids of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

1, 1, 1, 6, 3, 1, 44, 16, 5, 1, 344, 116, 30, 7, 1, 2856, 928, 224, 48, 9, 1, 24816, 7856, 1840, 376, 70, 11, 1, 223016, 69264, 15912, 3184, 580, 96, 13, 1, 2056256, 629472, 142592, 28176, 5080, 844, 126, 15, 1, 19344472, 5855472, 1312360, 256992, 46072

Offset: 0

Author

Emeric Deutsch, Jun 11 2005

Keywords

• nonn
• tabl

Comments

Also 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 pyramids of the second kind (a pyramid of the second kind is a sequence U^pd^(2p) for some positive integer p, starting at the x-axis). Row sums yield A027307. Column 0 yields A108452. Number of pyramids of the first kind in all paths from (0,0) to (3n,0) is given by A108453.

Examples

			T(2,1)=3 because we have (ud)Udd, (uudd) and Udd(ud), the pyramids of the first kind being shown between parentheses.
Triangle begins:
1;
1,1;
6,3,1;
44,16,5,1;
		

Links

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

Crossrefs

Cf. A027307, A108452, A108453, A108445.

Programs

Maple

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=(1-z)/(1-t*z-z*(1-z)*A*(1+A)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,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

Formula

G.f.: (1-z)/[1-tz-z(1-z)A(1+A)], 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).