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A051288
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Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UUDs.
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0
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1, 2, 4, 2, 8, 12, 16, 48, 6, 32, 160, 60, 64, 480, 360, 20, 128, 1344, 1680, 280, 256, 3584, 6720, 2240, 70, 512, 9216, 24192, 13440, 1260, 1024, 23040, 80640, 67200, 12600, 252, 2048, 56320, 253440, 295680, 92400, 5544, 4096, 135168, 760320
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| By reading paths backward, the UUD in the name could be replaced by DDU.
Or, triangular array T read by rows: T(n,k)=P(2n,n,4k), where P(n,k,c)=number of vectors (x(1),x(2,),...,x(n)) of k 1's and n-k 0's such that x(i)=x(n+1-i) for exactly c values of i. P(n,k,n) counts palindromes.
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FORMULA
| a(n, k)=binom(n, 2k)2^(n-2k)binom(2k, k). G.f. (1-4x+4x^2(1-y))^(-1/2) = Sum_{n>=0, k>=0} a(n, k) x^n y^k.
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EXAMPLE
| Table begins
\ k 0, 1, 2, ...
n
0 | 1
1 | 2
2 | 4, 2
3 | 8, 12,
4 | 16, 48, 6
5 | 32, 160, 60
6 | 64, 480, 360, 20
7 |128, 1344, 1680, 280
a(2,1)=2 because UUDD, DUUD each have one UUD.
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MATHEMATICA
| Table[Binomial[n, 2k]2^(n-2k)Binomial[2k, k], {n, 0, 15}, {k, 0, n/2}]
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CROSSREFS
| Row sums are the (even) central binomial coefficients A000984. A091894 gives the distribution of the parameter "number of DDUs" on Dyck paths.
Sequence in context: A068217 A114593 A114655 * A120434 A187619 A008303
Adjacent sequences: A051285 A051286 A051287 * A051289 A051290 A051291
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KEYWORD
| nonn,tabf
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Additional comments from David Callan (callan(AT)stat.wisc.edu), Aug 28 2004
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