A288750 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals nine.

1, 1, 21, 98, 568, 2858, 14128, 67556, 316490, 1456952, 6612520, 29652948, 131613716, 578987886, 2527351698, 10956840549, 47212399022, 202328867061, 862840720214, 3663367687951, 15491222396862, 65268041732681, 274068630138339, 1147305286307251

Offset: 9

Author

Alois P. Heinz, Jun 14 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..1000
• Wikipedia, Counting lattice paths

Crossrefs

Column k=9 of A287822.

Cf. A000108.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 9)-g(n, 8):
seq(a(n), n=9..35);

Mathematica

b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 9] - g[n, 8], {n, 9, 35}] (* Indranil Ghosh, Aug 08 2017 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
def a(n): return g(n, 9) - g(n, 8)
print([a(n) for n in range(9, 36)]) # Indranil Ghosh, Aug 08 2017