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%I #11 Jul 23 2017 12:17:29
%S 1,1,2,1,7,2,1,18,15,2,1,41,68,25,2,1,88,244,171,37,2,1,183,765,866,
%T 351,51,2,1,374,2199,3651,2355,636,67,2,1,757,5954,13601,12708,5421,
%U 1058,85,2,1,1524,15438,46355,58977,36198,11116,1653,105,2,1,3059,38747,147768
%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k peaks of height >1 (n >= 1; 0 <= k <= n-1).
%C 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 Row sums yield A002212.
%H 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 T(n,0) = 1.
%F Sum_{k=0..n-1} k*T(n,k) = A128748(n).
%F G.f.: G(t,z) = (1 - z + z*K(t,z))/(1 - z*K(t,z)) - 1, where K = K(t,z) satisfies zK^2 - (1 - tz)K + 1 - z = 0 (K is the g.f. for the number of peaks; see A126182).
%e T(3,1)=7 because we have UDU(UD)D, UDU(UD)L, U(UD)DUD, UU(UD)DD, UU(UD)LD, UU(UD)DL and UU(UD)LL (the peaks of height >1 are shown between parentheses).
%e Triangle starts:
%e 1;
%e 1, 2;
%e 1, 7, 2;
%e 1, 18, 15, 2;
%e 1, 41, 68, 25, 2;
%p K:=(1-z*t-sqrt(z^2*t^2-2*z*t+1+4*z^2-4*z))/2/z: G:=z*(2*K-1)/(1-z*K): Gser:=simplify(series(G,z=0,14)): 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=0..n-1) od; # yields sequence in triangular form
%Y Cf. A002212, A128748.
%K tabl,nonn
%O 1,3
%A _Emeric Deutsch_, Mar 31 2007
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