A129157 - OEIS

%I #15 Dec 19 2015 18:03:50

%S 1,1,2,1,5,5,14,22,42,94,1,132,400,11,429,1709,81,1430,7351,503,1,

%T 4862,31857,2851,17,16796,139100,15297,176,58786,611781,79228,1440,1,

%U 208012,2709230,400694,10259,23,742900,12075248,1993226,66774,307,2674440

%N 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 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 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 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 Row n has 1+floor((n+1)/3) terms (n>=1).

%F Row sums yield A002212.

%F T(n,0) = binomial(2*n,n)/(n+1) = A000108(n) (the Catalan numbers).

%F Sum_{k>=0} k*T(n,k) = A129158(n).

%e T(3,1) = 5 because we have UD(UUDL), (UUUDLD), (UUDUDL), (UUUDDL) and (UUUDLL);

%e T(5,2) = 1 because we have (UUUDLD)(UUDL) (the primitive non-Dyck factors are shown between parentheses).

%e Triangle starts:

%e     1;

%e     1;

%e     2,   1;

%e     5,   5;

%e    14,  22;

%e    42,  94,  1;

%e   132, 400, 11;

%p 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 Gser:=simplify(series(G,z=0,18)):

%p for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od:

%p for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..floor((n+1)/3)) od;

%p # yields sequence in triangular form

%Y Cf. A000108, A002212, A129154, A129158.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Apr 02 2007