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A097692
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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 UDUs.
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2
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1, 2, 4, 2, 10, 8, 2, 26, 30, 12, 2, 70, 104, 60, 16, 2, 192, 350, 260, 100, 20, 2, 534, 1152, 1050, 520, 150, 24, 2, 1500, 3738, 4032, 2450, 910, 210, 28, 2, 4246, 12000, 14952, 10752, 4900, 1456, 280, 32, 2
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OFFSET
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0,2
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COMMENTS
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See A091869 for the distribution of the parameter "number of UDUs" on Dyck paths.
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REFERENCES
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Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2. - From N. J. A. Sloane, Feb 06 2013
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LINKS
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Table of n, a(n) for n=0..45.
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FORMULA
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G.f. ((1 + x - x*y)/(1 - 3*x - x*y))^(1/2) = Sum_{n>=0, k>=0} a(n, k) x^n y^k.
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EXAMPLE
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Table begins
\ k 0, 1, 2, ...
n
0 | 1
1 | 2
2 | 4, 2
3 | 10, 8, 2
4 | 26, 30, 12, 2
5 | 70, 104, 60, 16, 2
6 |192, 350, 260, 100, 20, 2
7 |534, 1152, 1050, 520, 150, 24, 2
The path UDUDUD contains 2 UDUs and a(2,1) = 2 because each of UDUD, DUDU
contains one UDU.
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MATHEMATICA
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gfForBalancedByNumberUDU=Sqrt[(1 + x - x*y)/(1 - 3*x - x*y)]; Map[CoefficientList[ #, y]&, CoefficientList[Normal[Series[gfForBalancedByNumberUDU, {x, 0, 8}, {y, 0, 8}]], x]]
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CROSSREFS
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Column k=0 is A025565. The row sums are the (even) central binomial coefficients A000984.
Sequence in context: A086930 A099585 A097577 * A118920 A162982 A125755
Adjacent sequences: A097689 A097690 A097691 * A097693 A097694 A097695
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KEYWORD
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nonn,tabf
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AUTHOR
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David Callan, Aug 19 2004; corrected Jun 10 2005
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EXTENSIONS
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Keyword tabl changed to tabf by Michel Marcus, Apr 07 2013
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STATUS
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approved
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