

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
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OFFSET

1,3


COMMENTS



LINKS



FORMULA

G.f.=tz(1+t)/[1tzt^2z(1+t)zAzA^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)/(1t*zt^2*z(1+t)*z*Az*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



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



