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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). 5
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; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]].

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 * A135305 A114463 A135309

Adjacent sequences:  A114489 A114490 A114491 * A114493 A114494 A114495

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 01 2005

STATUS

approved

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Last modified April 24 22:26 EDT 2019. Contains 322446 sequences. (Running on oeis4.)