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A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level. 1
1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 24, 11, 5, 1, 1, 75, 35, 14, 6, 1, 1, 243, 113, 47, 17, 7, 1, 1, 808, 376, 156, 60, 20, 8, 1, 1, 2742, 1276, 532, 204, 74, 23, 9, 1, 1, 9458, 4402, 1840, 712, 257, 89, 26, 10, 1, 1, 33062, 15390, 6448, 2507, 917, 315, 105, 29, 11, 1, 1, 116868 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

T(n,0) = A000958(n-1). - Emeric Deutsch, Dec 23 2006

LINKS

Table of n, a(n) for n=1..67.

Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.

FORMULA

See Mathematica line.

G.f.=zC/(1+z-tz-zC), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, Dec 23 2006

With offset 0 (0<=k<=n), T(n,k)=A065600(n,k)+A065600(n+1,k)-A065600(n,k-1). - Philippe Deléham, Apr 01 2007

EXAMPLE

Triangle begins:

1

1 1

3 1 1

8 4 1 1

24 11 5 1 1

75 35 14 6 1 1

T(4,2)=1 because we have UDUDUUDD.

MAPLE

c:=(1-sqrt(1-4*z))/2/z: G:=z*c/(1-t*z+z-z*c): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 12 do seq(coeff(P[n], t, k), k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 23 2006

MATHEMATICA

u[n_, k_, i_]:=(2i+1)/(n-k)Binomial[k+i, i]Binomial[2n-2k-2i-2, n-k-1] u[n_, k_]/; k<=n-1 := Sum[u[n, k, i], {i, 0, n-k-1}] Table[u[n, k], {n, 10}, {k, 0, n-1}] (* u[n, k, i] is the number of Dyck n-paths with k low UDUs and k+i+1 returns altogether. For example, with n=4, k=1 and i=1, u[n, k, i] counts UDUUDDUD, UUDDUDUD because each has size n=4, k=1 low UDUs and k+i+1=3 returns to ground level. *) (* David Callan, Nov 03 2005 *)

CROSSREFS

Cf. A091869, A092107.

Cf. A000958.

Sequence in context: A143953 A114276 A152879 * A122897 A117425 A287215

Adjacent sequences:  A098744 A098745 A098746 * A098748 A098749 A098750

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Oct 30 2004

STATUS

approved

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Last modified May 28 07:57 EDT 2017. Contains 287212 sequences.