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%I #13 Jul 23 2017 12:17:58
%S 1,0,1,1,1,1,4,3,2,1,15,11,6,3,1,59,41,22,10,4,1,241,159,84,38,15,5,1,
%T 1011,639,331,150,60,21,6,1,4326,2640,1342,606,246,89,28,7,1,18797,
%U 11146,5570,2500,1023,380,126,36,8,1,82685,47884,23567,10503,4312,1630
%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive Dyck factors (n >= 0; 0 <= k <= n).
%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. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.
%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) 2192-2203.
%F T(n,0) = A129155(n).
%F Sum_{k=0..n} k*T(n,k) = A129156(n).
%F G.f.: G(t,z) = (1 + z(g-1))/(1 - z(g-C) - tzC), 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.
%e T(4,2)=6 because we have (UD)(UD)UUDL, (UD)(UUDUDD), (UD)(UUUDDD), (UUDD)(UUDD), (UUDUDD)(UD) and (UUUDDD)(UD) (the primitive Dyck factors are shown between parentheses).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 1, 1, 1;
%e 4, 3, 2, 1;
%e 15, 11, 6, 3, 1;
%p G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2-t+z+(t-1)*sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y Cf. A002212, A129155, A129156.
%K nonn,tabl
%O 0,7
%A _Emeric Deutsch_, Apr 02 2007
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