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

0,5

Comments

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

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.

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)

Example

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.

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 *)

Crossrefs

Row sums are the Catalan numbers A000108.

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

Sequence in context: A031111 A089911 A098978 * A247193 A362358 A096320

Adjacent sequences: A111298 A111299 A111300 * A111302 A111303 A111304

Keyword

nonn,tabf

Author

David Callan, Nov 02 2005

Status

approved