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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).

%I A128745 #10 Jul 23 2017 12:17:24
%S A128745 1,1,2,2,4,4,6,10,12,8,21,32,36,32,16,79,116,124,112,80,32,311,448,
%T A128745 468,416,320,192,64,1265,1800,1860,1640,1280,864,448,128,5275,7440,
%U A128745 7640,6720,5280,3712,2240,1024,256,22431,31426,32136,28256,22336,16032,10304
%N 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).
%C A128745 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.
%C A128745 Row sums yield A002212.
%C A128745 T(n,1) = A033321(n-1).
%C A128745 Sum_{k=1..n} k*T(n,k) = A128746(n).
%H A128745 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%F A128745 G.f.: t*z/(1 - z*g - 2*t*z), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
%e A128745 T(3,2)=4 because we have UDUUDD, UDUUDL, UUDUDD and UUDUDL.
%e A128745 Triangle starts:
%e A128745    1;
%e A128745    1,  2;
%e A128745    2,  4,  4;
%e A128745    6, 10, 12,  8;
%e A128745   21, 32, 36, 32, 16;
%p A128745 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
%Y A128745 Cf. A002212, A033321, A128746.
%K A128745 tabl,nonn
%O A128745 1,3
%A A128745 _Emeric Deutsch_, Mar 31 2007