OFFSET
0,7
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..100, flattened
Andrei Asinowski, Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
FORMULA
G.f.: c(z/(1+(1-t^2)z+(1-t)^2*z^2)), where c(z) = (1-sqrt(1-4z))*(2z) is the g.f. of the Catalan numbers (A000108).
The trivariate g.f., with z marking semilength, t marking number of UDU's and s marking number of DUD's is c(z/(1+(1-ts)*z + (1-t)(1-s)z^2)), where c(z) = (1-sqrt(1-4z))*(2z) is the g.f. of the Catalan numbers (A000108).
EXAMPLE
T(4,3) = 4 because we have UDUDUUDD, UUDDUDUD, UDUUDUDD and UUDUDDUD.
Triangle starts:
1;
1;
1, 0, 1;
1, 2, 1, 0, 1;
2, 4, 2, 4, 1, 0, 1;
4, 8, 10, 8, 4, 6, 1, 0, 1;
MAPLE
c := proc (z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G := c(z/(1+(1-t^2)*z+(1-t)^2*z^2)): Gser := simplify(series(G, z = 0, 12)): for n from 0 to 9 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 9 do seq(coeff(P[n], t, k), k = 0 .. 2*n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2, 4][t])*
`if`(t=3, z, 1)+b(x-1, y-1, [5, 3, 5, 3, 5][t])*`if`(t=4, z, 1))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=1..12); # Alois P. Heinz, Jun 04 2014
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 2, 4, 2, 4}[[t]]]*If[t == 3, z, 1] + b[x-1, y-1, {5, 3, 5, 3, 5}[[t]]]*If[t == 4, z, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 10 2008
STATUS
approved