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A111301
Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level.
3
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
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: A380820 A089911 A098978 * A247193 A362358 A096320
KEYWORD
nonn,tabf
AUTHOR
David Callan, Nov 02 2005
STATUS
approved