|
| |
|
|
A129183
|
|
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the sum of the height of the peaks is k (n>=0; n<=k<=floor((n+1)^2/4)).
|
|
2
|
|
|
|
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, 0, 0, 0, 0, 0, 0, 32, 32, 30, 20, 12, 4, 2, 0, 0, 0, 0, 0, 0, 0, 64, 80, 88, 73, 56, 34, 20, 9, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 128, 192, 240, 232, 206, 156, 116, 72, 46, 24, 12, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
REFERENCES
|
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 Math., 112, 1993, 65-79.
|
|
|
LINKS
|
Table of n, a(n) for n=0..87.
|
|
|
FORMULA
|
G.f.=G(t,z)=H(t,1,z), where H(t,x,z)=1+z[H(t,tx,z)-1+tx]H(t,x,z) (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
|
|
|
CROSSREFS
|
Cf. A000108, A011782, A000079, A000302.
Sequence in context: A100951 A190608 A011991 * A181566 A110173 A131427
Adjacent sequences: A129180 A129181 A129182 * A129184 A129185 A129186
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
Emeric Deutsch, Apr 07 2007
|
|
|
STATUS
|
approved
|
| |
|
|
