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 A128744 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the first peak equal to k (1 <= k <= n). 0
 1, 1, 2, 3, 3, 4, 10, 10, 8, 8, 36, 36, 29, 20, 16, 137, 137, 111, 78, 48, 32, 543, 543, 442, 315, 200, 112, 64, 2219, 2219, 1813, 1306, 848, 496, 256, 128, 9285, 9285, 7609, 5527, 3649, 2200, 1200, 576, 256, 39587, 39587, 32521, 23779, 15901, 9802, 5552, 2848 (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 the 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,1) = A002212(n-1). T(n,2) = A002212(n-1) for n >= 3. Sum_{k=1..n} k*T(n,k) = A039919(n+1). G.f.: t*z*g/(1 - t*z - t*z*g), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z). EXAMPLE T(3,3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL. Triangle starts:    1;    1,  2;    3,  3,  4;   10, 10,  8,  8;   36, 36, 29, 20, 16; MAPLE g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z*g/(1-t*z-t*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, A039919. Sequence in context: A227263 A111574 A173590 * A293984 A207608 A290818 Adjacent sequences:  A128741 A128742 A128743 * A128745 A128746 A128747 KEYWORD tabl,nonn AUTHOR Emeric Deutsch, Mar 31 2007 STATUS approved

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Last modified January 24 01:05 EST 2020. Contains 331178 sequences. (Running on oeis4.)