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 A128718 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UU's (doublerises) (n >= 1; 0 <= k <= n-1). 2
 1, 1, 2, 1, 5, 4, 1, 9, 18, 8, 1, 14, 50, 56, 16, 1, 20, 110, 220, 160, 32, 1, 27, 210, 645, 840, 432, 64, 1, 35, 364, 1575, 3150, 2912, 1120, 128, 1, 44, 588, 3388, 9534, 13552, 9408, 2816, 256, 1, 54, 900, 6636, 24822, 49644, 53088, 28800, 6912, 512, 1, 65, 1320, 12090, 57750, 153426, 231000, 193440, 84480, 16640, 1024 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. Row sums yield A002212. LINKS E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203. FORMULA T(n,0) = 1; T(n,1) = (n-1)(n+2)/2 = A000096(n-1); T(n,k) = A126182(n,n-k), i.e., triangle is mirror image of A126182. Sum_{k=0..n-1} k*T(n,k) = A128743(n). T(n,k) = (binomial(n,k)/n)*Sum_{j=0..k} binomial(k,j)*binomial(n-k+j, j+1) (1 <= k <= n). G.f.: G - 1, where G = G(t,z) satisfies G = 1 + tzG^2 + zG - tz. EXAMPLE T(3,2)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL. Triangle starts:   1;   1,  2;   1,  5,  4;   1,  9, 18,  8;   1, 14, 50, 56, 16; MAPLE T:=proc(n, k) if k=0 then 1 else binomial(n, k)*sum(binomial(k, j)*binomial(n-k+j, j+1), j=0..k)/n fi end: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form MATHEMATICA m = 12; G[_] = 0; Do[G[z_] = 1 + t z G[z]^2 + z G[z] - t z + O[z]^m, {m}]; CoefficientList[#, t]& /@ CoefficientList[G[z], z] // Rest // Flatten (* Jean-François Alcover, Nov 15 2019 *) CROSSREFS Cf. A000096, A002212, A126182, A128743. Sequence in context: A274105 A056242 A343960 * A112358 A126351 A157011 Adjacent sequences:  A128715 A128716 A128717 * A128719 A128720 A128721 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Mar 30 2007 STATUS approved

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Last modified October 4 06:38 EDT 2022. Contains 357237 sequences. (Running on oeis4.)