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A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs. 5
1, 2, 4, 2, 10, 8, 2, 26, 30, 12, 2, 70, 104, 60, 16, 2, 192, 350, 260, 100, 20, 2, 534, 1152, 1050, 520, 150, 24, 2, 1500, 3738, 4032, 2450, 910, 210, 28, 2, 4246, 12000, 14952, 10752, 4900, 1456, 280, 32, 2, 12092, 38214, 54000, 44856, 24192, 8820, 2184, 360, 36, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

See A091869 for the distribution of the parameter "number of UDUs" on Dyck paths.

REFERENCES

Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2. - From N. J. A. Sloane, Feb 06 2013

LINKS

Alois P. Heinz, Rows n = 0..141, flattened

FORMULA

G.f.: ((1 + x - x*y)/(1 - 3*x - x*y))^(1/2) = Sum_{n>=0, k>=0} a(n,k) x^n y^k.

EXAMPLE

Table begins

\ k 0, 1, 2, ...

n

0 | 1

1 | 2

2 | 4, 2

3 | 10, 8, 2

4 | 26, 30, 12, 2

5 | 70, 104, 60, 16, 2

6 |192, 350, 260, 100, 20, 2

7 |534, 1152, 1050, 520, 150, 24, 2

The path UDUDUD contains 2 UDUs and a(2,1) = 2 because each of UDUD, DUDU contains one UDU.

MAPLE

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1,

      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))

      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))

    end:

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

seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015

MATHEMATICA

gfForBalancedByNumberUDU=Sqrt[(1 + x - x*y)/(1 - 3*x - x*y)]; Map[CoefficientList[ #, y]&, CoefficientList[Normal[Series[gfForBalancedByNumberUDU, {x, 0, 8}, {y, 0, 8}]], x]]

CROSSREFS

Column k=0 is A025565. The row sums are the (even) central binomial coefficients A000984.

Cf. A171651.

Sequence in context: A099585 A236959 A097577 * A118920 A305260 A162982

Adjacent sequences:  A097689 A097690 A097691 * A097693 A097694 A097695

KEYWORD

nonn,tabf

AUTHOR

David Callan, Aug 19 2004; corrected Jun 10 2005

EXTENSIONS

Keyword tabl changed to tabf by Michel Marcus, Apr 07 2013

STATUS

approved

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Last modified May 20 20:15 EDT 2022. Contains 353876 sequences. (Running on oeis4.)