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A129157 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive non-Dyck factors (n>=0; 0<=k<=floor((n+1)/3)).

%I A129157 #15 Dec 19 2015 18:03:50
%S A129157 1,1,2,1,5,5,14,22,42,94,1,132,400,11,429,1709,81,1430,7351,503,1,
%T A129157 4862,31857,2851,17,16796,139100,15297,176,58786,611781,79228,1440,1,
%U A129157 208012,2709230,400694,10259,23,742900,12075248,1993226,66774,307,2674440
%N A129157 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive non-Dyck factors (n>=0; 0<=k<=floor((n+1)/3)).
%C A129157 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. A primitive non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path.
%H A129157 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) 2192-2203
%F A129157 G.f.: G(t,z) = [1+tz(g-1)]/[1-tz(g-C)-zC], where g=1+zg^2+z(g-1) = [1-z-sqrt(1- 6z+5z^2)]/(2z) and C=1+zC^2=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%F A129157 Row n has 1+floor((n+1)/3) terms (n>=1).
%F A129157 Row sums yield A002212.
%F A129157 T(n,0) = binomial(2*n,n)/(n+1) = A000108(n) (the Catalan numbers).
%F A129157 Sum_{k>=0} k*T(n,k) = A129158(n).
%e A129157 T(3,1) = 5 because we have UD(UUDL), (UUUDLD), (UUDUDL), (UUUDDL) and (UUUDLL);
%e A129157 T(5,2) = 1 because we have (UUUDLD)(UUDL) (the primitive non-Dyck factors are shown between parentheses).
%e A129157 Triangle starts:
%e A129157     1;
%e A129157     1;
%e A129157     2,   1;
%e A129157     5,   5;
%e A129157    14,  22;
%e A129157    42,  94,  1;
%e A129157   132, 400, 11;
%p A129157 G:=(2+t-3*t*z-t*sqrt(1-6*z+5*z^2))/(1+t*z+(1-t)*sqrt(1-4*z)+t*sqrt(1-6*z+5*z^2)):
%p A129157 Gser:=simplify(series(G,z=0,18)):
%p A129157 for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od:
%p A129157 for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..floor((n+1)/3)) od;
%p A129157 # yields sequence in triangular form
%Y A129157 Cf. A000108, A002212, A129154, A129158.
%K A129157 nonn,tabf
%O A129157 0,3
%A A129157 _Emeric Deutsch_, Apr 02 2007