A108445 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 (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).
1, 0, 2, 4, 2, 4, 32, 18, 8, 8, 252, 146, 60, 24, 16, 2112, 1186, 496, 176, 64, 32, 18484, 10146, 4148, 1488, 480, 160, 64, 166976, 90162, 36216, 12792, 4160, 1248, 384, 128, 1545548, 824114, 326828, 113960, 36720, 11104, 3136, 896, 256, 14583808, 7699394
Offset: 0
Examples
T(2,1)=2 because we have uudd and UUdddd. Triangle begins: 1; 0,2; 4,2,4; 32,18,8,8; 252,146,60,24,16;
Links
- Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
Programs
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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+z-2*t*z-z*(1-z)*A*(1+A)): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 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+z-2tz-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).
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