OFFSET
0,6
COMMENTS
Row n has 1+floor((n+1)^2/4) terms, the first n of which are equal to 0. Row sums yield the Catalan numbers (A000108). T(n,n) = 2^(n-1) = A011782(n) = A000079(n-1) for n>=1. Sum(k*T(n,k), k>=0) = 4^(n-1) = A000302(n-1).
Also number of parallelogram polyominoes of semiperimeter n+1 and having area equal to k. Example: T(3,4)=1 because the square with side 2 is the only parallelogram polyomino with semiperimeter 4 and area 4. - Emeric Deutsch, Apr 07 2007
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
M. P. Delest and J. M. Fedou, Counting polyominoes using attribute grammars, Lecture Notes in Computer Science, vol. 461, pp. 46-60, Springer, Berlin, 1990.
M. P. Delest and J. M. Fedou, Attribute grammars are useful for combinatorics, Theor. Comp. Sci., 98, 1992, 65-76.
M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics. vol.112, no.1-3, pp. 65-79, (1993).
FORMULA
G.f.: G(t,z)=H(t,1,z), where H(t,x,z)=1+z*(H(t,t*x,z)-1+t*x)*H(t,x,z) where H(t,x,z) is the trivariate g.f. for Dyck paths according to sum of the height of the peaks, number of peaks and semilength, marked by t,x and z, respectively.
EXAMPLE
T(4,5) = 4 because we have UDUUDUDD, UUDUDDUD, UUDUUDDD and UUUDDUDD.
Triangle starts:
1;
0,1;
0,0,2;
0,0,0,4,1;
0,0,0,0,8,4,2;
0,0,0,0,0,16,12,9,4,1;
MAPLE
H:=1/(1-z*h[1]+z-z*t*x): for n from 1 to 11 do h[n]:=1/(1-z*h[n+1]+z-z*t^(n+1)*x) od: h[12]:=0: x:=1: G:=simplify(H): Gser:=simplify(series(G, z=0, 11)): for n from 0 to 9 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 9 do seq(coeff(P[n], t, j), j=0..floor((n+1)^2/4)) od; # 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, 1)+ `if`(t=1, z^y, 1)*b(x-1, y-1, 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..10); # Alois P. Heinz, Jun 10 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, 1] + If[t == 1, z^y, 1]*b[x-1, y-1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 07 2007
STATUS
approved