The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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: A194682 A274105 A056242 * A112358 A126351 A157011 Adjacent sequences:  A128715 A128716 A128717 * A128719 A128720 A128721 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Mar 30 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 30 18:38 EDT 2020. Contains 334728 sequences. (Running on oeis4.)