OFFSET
1,3
LINKS
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
FORMULA
G.f.=G=G(t, z)=1/(1-tzA-t^2*zA^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).
EXAMPLE
T(2,3)=2 because we have uUddd and Uuddd.
Triangle begins:
1,1;
2,5,2,1;
10,28,13,11,3,1;
66,196,90,89,34,18,4,1;
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*z*A-t^2*z*A^2)-1: Gserz:=simplify(series(G, z=0, 10)): for n from 1 to 9 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 1 to 9 do seq(coeff(P[n], t^k), k=1..2*n) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 04 2005
STATUS
approved