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A116424 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDUU's, 0<=k<=[(n-1)/2]. 3
1, 1, 2, 4, 1, 9, 5, 22, 19, 1, 57, 66, 9, 154, 221, 53, 1, 429, 729, 258, 14, 1223, 2391, 1131, 116, 1, 3550, 7829, 4652, 745, 20, 10455, 25638, 18357, 4115, 220, 1, 31160, 84033, 70404, 20598, 1790, 27, 93802, 275765, 264563, 96286, 12104, 379, 1, 284789 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

T(n,k) also gives the number of Dyck paths of semilength n with k UUDU's.

Column k=0 gives A105633(n-1) for n>0.

LINKS

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

Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

FORMULA

T(n,k) = Sum((-1)^(i+k) * binomial(i,k) * binomial(n-i,i) * binomial(2*n-3*i, n - 2*i -1)/(n-i), i=k..[(n-1)/2]), n >=1.

G.f. G = G(t,z) satisfies G = 1 + z^2(1-t)G + z(1-z+tz)G^2.

EXAMPLE

Triangle begins:

00 :     1;

01 :     1;

02 :     2;

03 :     4,    1;

04 :     9,    5;

05 :    22,   19,    1;

06 :    57,   66,    9;

07 :   154,  221,   53,   1;

08 :   429,  729,  258,  14;

09 :  1223, 2391, 1131, 116,  1;

10 :  3550, 7829, 4652, 745, 20;

...

T(4,1) = 5 because there exist five Dyck paths of semilength 4 with one occurrence of UDUU : UDUUUDDD, UDUUDUDD, UDUUDDUD, UUDUUDDD, UDUDUUDD.

MAPLE

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,

     `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])*

     `if`(t=4, z, 1) +b(x-1, y-1, [1, 3, 1, 3][t]))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):

seq(T(n), n=0..15);  # Alois P. Heinz, Jun 02 2014

MATHEMATICA

Series[((1 + (t - 1)z^2) - Sqrt[(1 + (t - 1)z^2)^2 - 4*z*(1 - z + z*t)])/(2* z*(1 - z + z*t)), {z, 0, 20}, {t, 0, 20}]

CROSSREFS

Cf. A105633, A243752.

Sequence in context: A273896 A163240 A091958 * A135306 A242352 A270953

Adjacent sequences:  A116421 A116422 A116423 * A116425 A116426 A116427

KEYWORD

nonn,tabf

AUTHOR

I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006

STATUS

approved

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Last modified May 23 01:01 EDT 2017. Contains 286908 sequences.