A111301 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level.

1, 1, 1, 1, 2, 3, 5, 8, 1, 14, 23, 5, 42, 70, 19, 1, 132, 222, 68, 7, 429, 726, 240, 34, 1, 1430, 2431, 847, 145, 9, 4862, 8294, 3003, 583, 53, 1, 16796, 28730, 10712, 2275, 262, 11, 58786, 100776, 38454, 8736, 1183, 76, 1, 208012, 357238, 138890, 33252, 5068

Offset: 0

Author

David Callan, Nov 02 2005

Keywords

• nonn
• tabf

Comments

Column k is the sum of columns 2k and 2k+1 of A106566.

Examples

			Table begins
k: ..0....1....2....3....
n
0 |..1
1 |..1
2 |..1....1
3 |..2....3
4 |..5....8....1
5 |.14...23....5
6 |.42...70...19....1
7 |132..222...68....7
a(3,1)=3 because the Dyck 3-paths containing one even-length descent to ground level are UUDUDD, UDUUDD, UUDDUD.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened
• David Callan, The 136th manifestation of C_n , arXiv:math/0511010 [math.CO], 2005.

Crossrefs

Row sums are the Catalan numbers A000108.

A143949 considers odd-length descents to the ground level. - Emeric Deutsch, Oct 05 2008

Programs

Mathematica

TableForm[Table[k/(n-k)Binomial[2n-2k, n]+(2k+1)/(2n-2k-1)Binomial[2n-2k-1, n], {n, 10}, {k, 0, n/2}]]
Join[{1}, Table[k/(n - k) Binomial[2 n - 2 k, n] + (2 k + 1)/(2 n - 2 k - 1) Binomial[2 n - 2 k - 1, n], {n, 25}, {k, 0, n/2}] // Flatten] (* G. C. Greubel, Jul 28 2017 *)

Formula

See Mathematica line.

From Emeric Deutsch, Oct 05 2008: (Start)

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

The trivariate g.f. H(t,s,z), where t (s) marks odd-length (even-length) descents to ground level and z marks semilength, is H=1/[1-z(t+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. (End)