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

9,3

Links

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Wikipedia, Counting lattice paths

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 *)

Prog

(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

Crossrefs

Column k=9 of A287822.

Cf. A000108.

Sequence in context: A264239 A200255 A212406 * A178794 A140370 A124949

Adjacent sequences: A288747 A288748 A288749 * A288751 A288752 A288753

Keyword

nonn

Author

Alois P. Heinz, Jun 14 2017

Status

approved