|
Search: id:A114492
|
|
|
| A114492 |
|
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k DDUU's, where U=(1,1), D=(1,-1) (0<=k<=floor(n/2)-1 for n>=2). |
|
+0
3
|
|
| 1, 1, 2, 5, 13, 1, 35, 7, 97, 34, 1, 275, 143, 11, 794, 558, 77, 1, 2327, 2083, 436, 16, 6905, 7559, 2180, 151, 1, 20705, 26913, 10051, 1095, 22, 62642, 94547, 43796, 6758, 268, 1, 190987, 328943, 183130, 37402, 2409, 29, 586219, 1136218, 742253, 191408
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Rows 0 and 1 contain one term each; row n contains floor(n/2) terms (n>=2). Row sums are the Catalan numbers (A000108). Column 0 yields A086581. Sum(k*T(n,k),k=0..floor(n/2)-1) = binomial(2n-3,n-4) (A003516).
|
|
REFERENCES
|
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
|
|
FORMULA
|
G.f. G=G(t, z) satisfies z(t+z-tz)G^2-(1-2(1-t)z+(1-t)z^2)G+1-z+tz=0.
|
|
EXAMPLE
|
T(5,1)=7 because we have UU(DDUU)DUDD, UU(DDUU)UDDD, UDUU(DDUU)DD, their mirror images, and UUU(DDUU)DDD (the DDUU's are shown between parentheses).
Triangle starts:
1;
1;
2;
5;
13,1;
35,7;
97,34,1;
|
|
MAPLE
|
G:=1/2/(-t*z-z^2+z^2*t)*(-1+2*z-2*t*z-z^2+z^2*t+sqrt(1+z^4-2*z^4*t+z^4*t^2-4*z+2*z^2-2*z^2*t)): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: 1; 1; for n from 0 to 14 do seq(coeff(t*P[n], t^j), j=1..floor(n/2)) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A000108, A086581, A003516.
Sequence in context: A137918 A114502 A135308 this_sequence A135305 A114463 A135309
Adjacent sequences: A114489 A114490 A114491 this_sequence A114493 A114494 A114495
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 01 2005
|
|
|
Search completed in 0.002 seconds
|
