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Search: id:A111301
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| A111301 |
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Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level. |
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3
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| 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
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Column k is the sum of columns 2k and 2k+1 of A106566.
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LINKS
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David Callan, The 136th manifestation of C_n .
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FORMULA
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See Mathematica line.
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), 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)
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EXAMPLE
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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.
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MATHEMATICA
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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}]]
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CROSSREFS
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Row sums are the Catalan numbers A000108.
A143949 considers odd-length descents to the ground level. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 05 2008]
Sequence in context: A031111 A089911 A098978 this_sequence A096320 A105955 A003893
Adjacent sequences: A111298 A111299 A111300 this_sequence A111302 A111303 A111304
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KEYWORD
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nonn,tabf
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AUTHOR
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David Callan (callan(AT)stat.wisc.edu), Nov 02 2005
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