login
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.
1

%I #3 Mar 30 2012 17:36:12

%S 1,0,1,0,1,1,1,1,2,1,5,2,3,3,1,19,7,5,6,4,1,67,26,12,11,10,5,1,232,93,

%T 38,23,21,15,6,1,804,325,131,61,44,36,21,7,1,2806,1129,456,192,105,80,

%U 57,28,8,1,9878,3935,1585,648,297,185,137,85,36,9,1,35072,13813,5520

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.

%C Row sums yield the Catalan numbers (A000108). T(n,0)=A114277(n-3) for n>=3. Sum(k*T(n,k), k=0..n)=A014318(n-1) for n>=1.

%F G.f.=G(t,z)=(1-2z)C/(1-z-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,k)=T(n-1,k)+T(n-1,k-1) for n,k>=1.

%e T(5,2)=5 because we have (UDUD)UUDUDD, (UUDDUUUDDD), (UUUDDDUUDD), (UDUUUUDDDD) and (UUUUDDDDUD) (the initial 2 pyramids are shown between parentheses).

%e Triangle starts:

%e 1;

%e 0,1;

%e 0,1,1;

%e 1,1,2,1;

%e 5,2,3,3,1;

%e 19,7,5,6,4,1;

%p C:=(1-sqrt(1-4*z))/2/z: G:=(1-2*z)*C/(1-z-t*z): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gser,z^n))) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

%Y Cf. A000108, A114277, A014318.

%K nonn,tabl

%O 0,9

%A _Emeric Deutsch_, Feb 27 2007