%I A118964
%S A118964 2,5,1,14,5,1,42,19,8,1,132,67,40,12,1,429,232,166,79,17,1,1430,804,634,
%T A118964 395,145,23,1,4862,2806,2335,1708,879,249,30,1,16796,9878,8480,6824,
%U A118964 4376,1823,404,38,1,58786,35072,30691,26137,19334,10521,3542,625,47,1
%N A118964 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength 
               n that have k double rises above the x-axis (n>=1,k>=0). (A Grand 
               Dyck path of semilength n is a path in the half-plane x>=0, starting 
               at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,
               -1); a double rise in a Grand Dyck path is an occurrence of uu in 
               the path.)
%C A118964 Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) 
               (the Catalan numbers). T(n,1)=A114277(n-2). Sum(k*T(n,k),k>=0)=A000531(n-1). 
               For all double rises (above, below and on the x-axis), see A118963.
%F A118964 G.f.=G(t,z)=(1+r)/[1-z(1+r)C]-1, where r=r(t,z) is the Narayana function, 
               defined by (1+r)(1+tr)z=r, r(t,0)=0 and C=C(z)=[1-sqrt(1-4z)]/(2z) 
               is the Catalan function. More generally, the g.f. H=H(t,s,u,z), where 
               t,s and u mark double rises above, below and on the x-axis, respectively, 
               is H=[1+r(s,z)]/[1-z(1+tr(t,z))(1+ur(s,z))].
%e A118964 T(3,1)=5 because we have u/ududd,u/uddud,udu/udd,duu/udd and u/udddu 
               (the double rises above the x-axis are indicated by /.
%e A118964 Triangle starts:
%e A118964 2;
%e A118964 5,1;
%e A118964 14,5,1;
%e A118964 42,19,8,1;
%e A118964 132,67,40,12,1;
%p A118964 C:=(1-sqrt(1-4*z))/2/z: r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/
               2/t/z: G:=(1+r)/(1-z*C*(1+r))-1: Gser:=simplify(series(G,z=0,15)): 
               for n from 1 to 11 do P[n]:=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 A118964 Cf. A000984, A000108, A114277, A000531, A118963.
%Y A118964 Sequence in context: A156067 A101920 A114494 this_sequence A073187 A138159 
               A118919
%Y A118964 Adjacent sequences: A118961 A118962 A118963 this_sequence A118965 A118966 
               A118967
%K A118964 nonn,tabf
%O A118964 1,1
%A A118964 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 07 2006