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A114486 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UUDD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=floor(n/2)). 1
1, 1, 1, 1, 3, 2, 10, 3, 1, 31, 8, 3, 98, 27, 6, 1, 321, 88, 16, 4, 1078, 287, 54, 10, 1, 3686, 960, 183, 28, 5, 12789, 3280, 616, 95, 15, 1, 44919, 11378, 2106, 332, 45, 6, 159407, 39953, 7323, 1152, 155, 21, 1, 570704, 141752, 25785, 4028, 556, 68, 7, 2058817 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(k*T(n,k),k=0..floor(n/2))=A000108(n-1) (the Catalan numbers). Column 0 yields A114487.

LINKS

Table of n, a(n) for n=0..56.

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 G=1+z(C-z+tz)G, where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. G=2/[1+2z^2-2tz^2+sqrt(1-4z)].

EXAMPLE

T(5,2)=3 because we have UUDDUUDDUD, UUDDUDUUDD and UDUUDDUUDD, where U=(1,1), D=(1,-1).

Triangle starts:

1;

1;

1,1;

3,2;

10,3,1;

31,8,3;

98,27,6,1; ...

MAPLE

C:=(1-sqrt(1-4*z))/2/z: eq:=G=1+z*(C-z+t*z)*G: G:=solve(eq, G): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 15 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form

CROSSREFS

Cf. A000108, A114487.

Sequence in context: A092502 A292923 A135515 * A176743 A220466 A090780

Adjacent sequences:  A114483 A114484 A114485 * A114487 A114488 A114489

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Nov 30 2005

STATUS

approved

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Last modified May 19 20:41 EDT 2019. Contains 323410 sequences. (Running on oeis4.)