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 A128724 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k LL's (n >= 0; 0 <= k <= n-2 for n >= 2). 3
 1, 1, 3, 9, 1, 30, 5, 1, 107, 23, 6, 1, 399, 105, 31, 7, 1, 1537, 477, 156, 40, 8, 1, 6069, 2166, 771, 219, 50, 9, 1, 24434, 9849, 3772, 1165, 295, 61, 10, 1, 99924, 44869, 18347, 6083, 1676, 385, 73, 11, 1, 413943, 204820, 88908, 31376, 9278, 2322, 490, 86, 12, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 steps in it. Row 0 and row 1 have one term each; row n has n-1 terms (n >= 2). Row sums yield A002212. LINKS Alois P. Heinz, Rows n = 0..150, flattened E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203 FORMULA T(n,0) = A128725(n). Sum_{k=0..n-2} k*T(n,k) = A128726(n). G.f.: G = G(t,z) satisfies z^2*G^3 - z(2 - tz)G^2 + (1 + z - z^2 - tzG + tz - 1 = 0. EXAMPLE T(4,1)=5 because we have UDUUUDLL, UUUUDLLD, UUDUUDLL, UUUDUDLL and UUUUDDLL. Triangle starts:     1;     1;     3;     9,  1;    30,  5,  1;   107, 23,  6,  1; MAPLE eq:=z^2*G^3-z*(2-t*z)*G^2+(1+z-z^2-t*z)*G+t*z-1=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form # second Maple program: b:= proc(n, y, t) option remember; expand(`if`(y>n, 0, `if`(n=0, 1,       `if`(t<0, 0, b(n-1, y+1, 1))+`if`(y<1, 0, b(n-1, y-1, 0))+       `if`(t>0 or y<1, 0, b(n-1, y-1, -1)*`if`(t<0, z, 1)))))     end: T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0\$2)): seq(T(n), n=0..14);  # Alois P. Heinz, Jun 19 2016 MATHEMATICA b[n_, y_, t_] := b[n, y, t] = Expand[If[y > n, 0, If[n == 0, 1, If[t < 0, 0, b[n - 1, y + 1, 1]] + If[y < 1, 0, b[n - 1, y - 1, 0]] + If[t > 0 || y < 1, 0, b[n - 1, y - 1, -1]*If[t < 0, z, 1]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *) CROSSREFS Cf. A002212, A128725, A128726. Sequence in context: A126177 A158483 A128733 * A128753 A179430 A016048 Adjacent sequences:  A128721 A128722 A128723 * A128725 A128726 A128727 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Mar 31 2007 STATUS approved

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Last modified April 7 01:11 EDT 2020. Contains 333291 sequences. (Running on oeis4.)