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 A128749 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1. 1
 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 14, 12, 9, 0, 1, 44, 53, 25, 14, 0, 1, 150, 196, 132, 44, 20, 0, 1, 520, 777, 555, 269, 70, 27, 0, 1, 1850, 3064, 2486, 1260, 485, 104, 35, 0, 1, 6696, 12233, 10902, 6264, 2496, 804, 147, 44, 0, 1, 24602, 49096, 47955, 30108, 13600 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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. An ascent in a path is a maximal sequence of consecutive U 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) = A128750(n). Sum_{k=0..n} k*T(n,k) = A085362(n-1). G.f.: G = G(t,z) satisfies z(1 + z - tz)G^2 - (1 - tz + tz^2 - z^2)G + 1 - z = 0. EXAMPLE T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD and UUD(U)DL (the ascents of length 1 are shown between parentheses). Triangle starts:    1;    0,  1;    2,  0,  1;    4,  5,  0,  1;   14, 12,  9,  0,  1;   44, 53, 25, 14,  0,  1; MAPLE eq:=z*(1+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form CROSSREFS Cf. A002212, A085362, A128750. Sequence in context: A077909 A247126 A229223 * A106579 A287318 A173003 Adjacent sequences:  A128746 A128747 A128748 * A128750 A128751 A128752 KEYWORD tabl,nonn AUTHOR Emeric Deutsch, Mar 31 2007 STATUS approved

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Last modified June 17 22:17 EDT 2019. Contains 324200 sequences. (Running on oeis4.)