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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.
0
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
1,3
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.
LINKS
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
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).
EXAMPLE
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;
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
CROSSREFS
Cf. A027307.
Sequence in context: A322398 A341606 A388996 * A307500 A049245 A123547
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 21 2005
STATUS
approved