A120059 Triangle read by rows: T(n,k) is the number of Dyck n-paths (A000108) whose longest pyramid has size k.
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, 212, 85, 25, 7, 2, 1, 0, 275, 723, 307, 90, 25, 7, 2, 1, 0, 794, 2491, 1121, 330, 91, 25, 7, 2, 1, 0, 2327, 8654, 4129, 1225, 335, 91, 25, 7, 2, 1
Offset: 0
Examples
Table begins \ k..0....1....2....3....4....5....6....7 n 0 |..1 1 |..0....1 2 |..0....1....1 3 |..0....2....2....1 4 |..0....5....6....2....1 5 |..0...13...19....7....2....1 6 |..0...35...63...24....7....2....1 7 |..0...97..212...85...25....7....2....1 a(3,2)=2 because the Dyck 3-paths whose longest pyramid has size 2 are UUDDUD, UDUUDD.
Programs
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Mathematica
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
=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}]
Formula
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).
Comments