A288751 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals ten.

1, 1, 23, 113, 690, 3620, 18562, 91628, 441694, 2086758, 9699872, 44473357, 201530730, 904002266, 4019194153, 17730014882, 77671514151, 338156070316, 1464024164516, 6306519136902, 27042360481662, 115475401501264, 491223012388610, 2082320685801754

Offset

10,3

Links

Alois P. Heinz, Table of n, a(n) for n = 10..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, 10)-g(n, 9):
seq(a(n), n=10..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, 10] - g[n, 9], {n, 10, 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, 10) - g(n, 9)
print([a(n) for n in range(10, 36)]) # Indranil Ghosh, Aug 08 2017

Crossrefs

Column k=10 of A287822.

Cf. A000108.

Sequence in context: A142324 A337425 A233362 * A070024 A111943 A057877

Adjacent sequences: A288748 A288749 A288750 * A288752 A288753 A288754

Keyword

nonn

Author

Alois P. Heinz, Jun 14 2017

Status

approved