OFFSET
0,2
LINKS
Alois P. Heinz, Rows n = 0..100, flattened
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
FORMULA
G.f. G = G(t,z) satisfies G = 1 + z(t + z - tz)^2*G^3 + z(2-t)(t + z - tz)G^2 + 2z(1-t)G.
EXAMPLE
T(2,1) = 3 because we have uUddd, Uuddd and UdUddd.
Triangle begins:
1;
2;
6, 3, 1;
21, 24, 15, 5, 1;
80, 150, 145, 84, 31, 7, 1;
322, 857, 1145, 949, 528, 202, 53, 9, 1;
MAPLE
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
`if`(x=0, 1, b(x-1, y-1, min(2, t+1))*`if`(t=2, z, 1)+
b(x-1, y+2, 0)+b(x-2, y+1, 0))))
end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(3*n, 0$2)):
seq(T(n), n=0..8); # Alois P. Heinz, Oct 06 2015
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, Min[2, t + 1]]*If[t == 2, z, 1] + b[x - 1, y + 2, 0] + b[x - 2, y + 1, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[3*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
PROG
(PARI) {T(n, k)=local(G=1+z*O(z^n)+t*O(t^k)); for(k=1, n, G=1+z*(t+z-t*z)^2*G^3+z*(2-t)*(t+z-t*z)*G^2+2*z*(1-t)*G); polcoeff(polcoeff(G, n, z), k, t)}
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch and Paul D. Hanna, Jun 10 2005
STATUS
approved