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 A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs. 5
 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, 12092, 38214, 54000, 44856, 24192, 8820, 2184, 360, 36, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See A091869 for the distribution of the parameter "number of UDUs" on Dyck paths. REFERENCES 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 LINKS Alois P. Heinz, Rows n = 0..141, flattened FORMULA 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. EXAMPLE 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. MAPLE b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1, expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1)) +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1))))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2, 1)): seq(T(n), n=0..12); # Alois P. Heinz, Apr 29 2015 MATHEMATICA 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]] CROSSREFS Column k=0 is A025565. The row sums are the (even) central binomial coefficients A000984. Cf. A171651. Sequence in context: A236959 A366032 A097577 * A118920 A305260 A162982 Adjacent sequences: A097689 A097690 A097691 * A097693 A097694 A097695 KEYWORD nonn,tabf AUTHOR David Callan, Aug 19 2004; corrected Jun 10 2005 EXTENSIONS Keyword tabl changed to tabf by Michel Marcus, Apr 07 2013 STATUS approved

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Last modified August 6 03:16 EDT 2024. Contains 374957 sequences. (Running on oeis4.)