A288746 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals five.

1, 1, 13, 48, 220, 925, 3895, 16137, 66399, 271446, 1101626, 4442143, 17822176, 71191082, 283269813, 1123212251, 4439583152, 17496345670, 68765995160, 269595218881, 1054499461385, 4115767918639, 16032123369549, 62333852291879, 241935803355457, 937486479689517

Offset

5,3

Links

Alois P. Heinz, Table of n, a(n) for n = 5..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, 5)-g(n, 4):
seq(a(n), n=5..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, 5] - g[n, 4], {n, 5, 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, 5) - g(n, 4)
print([a(n) for n in range(5, 36)]) # Indranil Ghosh, Aug 08 2017

Crossrefs

Column k=5 of A287822.

Cf. A000108.

Sequence in context: A225920 A027980 A200254 * A220707 A189349 A013200

Adjacent sequences: A288743 A288744 A288745 * A288747 A288748 A288749

Keyword

nonn

Author

Alois P. Heinz, Jun 14 2017

Status

approved