OFFSET
0,3
COMMENTS
FORMULA
G.f.: G=G(t,z) is given by G = C + z*C*(t*(G - 1 - z - z^2*G) + 1 + z + z^2*G), where C=C(t,z) is the solution of the equation z^2*(t+z^2-t*z^2)*C^2 - (1 - z^2 + t*z^2)*C + 1 = 0.
EXAMPLE
T(6,2)=3 because we have UD(U[UU)U], (U[UU)U]DD, and (U[UU)U]DU, where U=(1,1) and D=(1,-1) (the triple-rises are shown between parentheses).
Triangle starts:
1;
1;
2;
2,1;
4,1,1;
5,3,1,1;
9,6,3,1,1;
MAPLE
eqC := z^2*(t+z^2-t*z^2)*C^2-(1-z^2+t*z^2)*C+1 = 0: C := RootOf(eqC, C): G := C*(1+z+z^2-t*z-t*z^2)/(1-t*z*C-z^3*C+t*z^3*C): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 13 do seq(coeff(P[n], t, k), k = 0 .. n-2) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 18 2011
STATUS
approved