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
Examples
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;
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/(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
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).
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