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A129170 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the sum of the height of the peaks is k (n >= 0; n <= k <= floor((n+1)^2/4)).

%I A129170 #7 Jul 23 2017 11:28:59
%S A129170 1,0,1,0,0,3,0,0,0,8,2,0,0,0,0,21,9,6,0,0,0,0,0,55,33,29,16,4,0,0,0,0,
%T A129170 0,0,144,111,114,84,60,18,12,0,0,0,0,0,0,0,377,355,409,356,305,199,
%U A129170 120,58,32,8,0,0,0,0,0,0,0,0,987,1098,1389,1365,1308,1032,843,507,372,204,120
%N A129170 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the sum of the height of the peaks is k (n >= 0; n <= k <= floor((n+1)^2/4)).
%C A129170 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 A129170 Row n has 1 + floor((n+1)^2/4) terms, the first n of which are equal to 0.
%C A129170 Row sums yield A002212.
%H A129170 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 A129170 T(n,n) = A088305(n) = A001906(n) = Fibonacci(2n) for n >= 1.
%F A129170 Sum_{k>=0} k*T(n,k) = A129171(n).
%F A129170 G.f.: G(t,z)=H(t,1,z), where H(t,x,z)=1+z[H(t,tx,z)-1+tx]H(t,x,z)+z[H(t,tx,z)-1] (H(t,x,z) is the trivariate g.f. for skew Dyck paths according to sum of the height of the peaks, number of peaks and semilength, marked by t,x and z, respectively).
%e A129170 T(3,4)=2 because we have UUDUDD and UUDUDL.
%e A129170 Triangle starts:
%e A129170   1;
%e A129170   0,  1;
%e A129170   0,  0,  3;
%e A129170   0,  0,  0,  8,  2;
%e A129170   0,  0,  0,  0, 21,  9,  6;
%e A129170   0,  0,  0,  0,  0, 55, 33, 29, 16,  4;
%p A129170 H:=(1+z*h[1]-z)/(1-z*h[1]+z-z*t*x): for n from 1 to 11 do h[n]:=(1+z*h[n+1]-z)/(1-z*h[n+1]+z-z*t^(n+1)*x) od: h[12]:=0: x:=1: G:=simplify(H): Gser:=simplify(series(G,z=0,11)): for n from 0 to 9 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 9 do seq(coeff(P[n],t,j),j=0..floor((n+1)^2/4)) od; # yields sequence in triangular form
%Y A129170 Cf. A001906, A002212, A088305, A129171.
%K A129170 nonn,tabf
%O A129170 0,6
%A A129170 _Emeric Deutsch_, Apr 07 2007