OFFSET
0,6
COMMENTS
FORMULA
G.f.: G=G(t,z) satisfies G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)), where c = (1-sqrt(1-4*z^2))/(2*z^2) (the Catalan function with argument z^2).
EXAMPLE
T(4,2)=2 because we have UDUD and UUDD, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
1;
1;
1, 1;
1, 2;
1, 3, 2;
1, 4, 5;
2, 5, 9, 4;
3, 6, 14, 12;
8, 9, 20, 25, 8;
MAPLE
eq := G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)): c := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := simplify(solve(eq, G)): gser := simplify(series(g, z = 0, 19)): for n from 0 to 15 do P[n] := sort(expand(coeff(gser, z, n))) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 04 2011
STATUS
approved