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A135309
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UUUU's starting at level 0.
1
1, 1, 2, 5, 13, 1, 36, 6, 105, 27, 319, 110, 1002, 427, 1, 3235, 1616, 11, 10685, 6034, 77, 35970, 22376, 440, 123045, 82725, 2241, 1, 426667, 305606, 10611, 16, 1496782, 1129683, 47823, 152, 5303623, 4181954, 208148, 1120
OFFSET
0,3
COMMENTS
From Emeric Deutsch, Dec 15 2007: (Start)
Row n has 1+floor(n/4) terms.
Row sums yield the Catalan numbers (A000108).
Column 0 yields A135310. (End)
LINKS
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
FORMULA
From Emeric Deutsch, Dec 15 2007: (Start)
T(n,k) = Sum_{j=k..floor(n/4)} (-1)^(j-k)*(5j+1)*binomial(j,k)*binomial(2n-3j, n+j)/(n+j+1).
G.f.: G(t,z) = C/(1+(1-t)*z^4*C^5), where C=(1-sqrt(1-4z))/(2z) is the g.f. of the Catalan numbers (A000108). (End)
EXAMPLE
Triangle begins:
1;
1;
2;
5;
13, 1;
36, 6,
105, 27,
319, 110,
1002, 427, 1;
3235, 1616, 11;
10685, 6034, 77;
...
T(5,1)=6 because we have UUUUUDDDDD, UUUUDUDDDD, UUUUDDUDDD, UUUUDDDUDD, UUUUDDDUDD and UUUUDDDDUD.
MAPLE
T:=proc(n, k) options operator, arrow: sum((-1)^(j-k)*(5*j+1)*binomial(j, k)*binomial(2*n-3*j, n+j)/(n+j+1), j=k..floor((1/4)*n)) end proc: for n from 0 to 15 do seq(T(n, k), k=0..floor((1/4)*n)) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 15 2007
CROSSREFS
Sequence in context: A114492 A135305 A114463 * A135331 A135329 A114508
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Dec 07 2007
EXTENSIONS
More terms from Emeric Deutsch, Dec 15 2007
STATUS
approved