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 A128745 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the last peak equal to k (1<=k<=n). 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 the path is defined to be the number of its steps. 1
 1, 1, 2, 2, 4, 4, 6, 10, 12, 8, 21, 32, 36, 32, 16, 79, 116, 124, 112, 80, 32, 311, 448, 468, 416, 320, 192, 64, 1265, 1800, 1860, 1640, 1280, 864, 448, 128, 5275, 7440, 7640, 6720, 5280, 3712, 2240, 1024, 256, 22431, 31426, 32136, 28256, 22336, 16032, 10304 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row sums yield A002212. T(n,1)=A033321(n-1). Sum(k*T(n,k),k=1..n)=A128746(n). LINKS E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203 FORMULA G.f.= tz/(1-zg-2tz), where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z). EXAMPLE T(3,2)=4 because we have UDUUDD, UDUUDL, UUDUDD and UUDUDL. Triangle starts: 1; 1,2; 2,4,4; 6,10,12,8; 21,32,36,32,16; MAPLE g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z/(1-2*t*z-z*g): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form CROSSREFS Cf. A002212, A033321, A128746. Sequence in context: A143483 A192326 A131733 * A188524 A126064 A066813 Adjacent sequences:  A128742 A128743 A128744 * A128746 A128747 A128748 KEYWORD tabl,nonn AUTHOR Emeric Deutsch, Mar 31 2007 STATUS approved

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