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A120059 Triangle read by rows: T(n,k) is the number of Dyck n-paths (A000108) whose longest pyramid has size k.

%I A120059 #2 Mar 31 2012 10:22:44
%S A120059 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 A120059 212,85,25,7,2,1,0,275,723,307,90,25,7,2,1,0,794,2491,1121,330,91,25,
%U A120059 7,2,1,0,2327,8654,4129,1225,335,91,25,7,2,1
%N A120059 Triangle read by rows: T(n,k) is the number of Dyck n-paths (A000108) whose longest pyramid has size k.
%C A120059 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 A120059 Generating function for column k>=1 is F[k]-F[k-1] where F[k]:=(1 + x^(k+1) - ((1 + x^(k+1))^2 - 4*x)^(1/2))/(2*x).
%e A120059 Table begins
%e A120059 \ k..0....1....2....3....4....5....6....7
%e A120059 n
%e A120059 0 |..1
%e A120059 1 |..0....1
%e A120059 2 |..0....1....1
%e A120059 3 |..0....2....2....1
%e A120059 4 |..0....5....6....2....1
%e A120059 5 |..0...13...19....7....2....1
%e A120059 6 |..0...35...63...24....7....2....1
%e A120059 7 |..0...97..212...85...25....7....2....1
%e A120059 a(3,2)=2 because the Dyck 3-paths whose longest pyramid has size 2 are
%e A120059 UUDDUD, UDUUDD.
%t A120059 Clear[a] (* a[n,k] is the number of Dyck n-paths 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[j-1,k] a[n-j,k],{j,n}] - a[n-k-1,k] Table[a[n,k]-a[n,k-1],{n,0,8},{k,0,n}]
%Y A120059 Cf. A120060. Column k=1 is A086581. Row sums are the Catalan numbers A000108.
%K A120059 nonn,tabl
%O A120059 0,8
%A A120059 _David Callan_, Jun 06 2006