%I
%S 1,0,1,0,1,1,0,2,2,1,0,5,6,2,1,0,13,19,7,2,1,0,35,63,24,7,2,1,0,97,
%T 212,85,25,7,2,1,0,275,723,307,90,25,7,2,1,0,794,2491,1121,330,91,25,
%U 7,2,1,0,2327,8654,4129,1225,335,91,25,7,2,1
%N Triangle read by rows: T(n,k) is the number of Dyck npaths (A000108) whose longest pyramid has size k.
%C A pyramid in a Dyck path is a subpath of the form U^k D^k with k>=1 (U=upstep, D=downstep). The longest pyramid is indicated by lowercase letters in the Dyck path UUDuuudddDUD and it has size 3.
%F Generating function for column k>=1 is F[k]F[k1] where F[k]:=(1 + x^(k+1)  ((1 + x^(k+1))^2  4*x)^(1/2))/(2*x).
%e Table begins
%e \ k..0....1....2....3....4....5....6....7
%e n
%e 0 ..1
%e 1 ..0....1
%e 2 ..0....1....1
%e 3 ..0....2....2....1
%e 4 ..0....5....6....2....1
%e 5 ..0...13...19....7....2....1
%e 6 ..0...35...63...24....7....2....1
%e 7 ..0...97..212...85...25....7....2....1
%e a(3,2)=2 because the Dyck 3paths whose longest pyramid has size 2 are
%e UUDDUD, UDUUDD.
%t Clear[a] (* a[n,k] is the number of Dyck npaths whose longest pyramid has size <=k *) a[0,k_]/;k>=0 := 1 a[1,k_]/;k>=1 := 1 a[n_,k_]/;k>=n := 1/(n+1)Binomial[2n,n] a[n_,0]/;n>=1 := 0 a[n_,k_]/;k<0:= 0 a[n_,k_]/; 1<=k && k<n && n>=2 := a[n,k] = Sum[a[j1,k] a[nj,k],{j,n}]  a[nk1,k] Table[a[n,k]a[n,k1],{n,0,8},{k,0,n}]
%Y Cf. A120060. Column k=1 is A086581. Row sums are the Catalan numbers A000108.
%K nonn,tabl
%O 0,8
%A _David Callan_, Jun 06 2006
