A288745 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals four.

1, 1, 11, 38, 163, 648, 2571, 10173, 40025, 156087, 605057, 2335566, 8980883, 34412583, 131431024, 500437733, 1900135511, 7196366668, 27191450135, 102522926104, 385785153584, 1448985664032, 5432879981201, 20337296148823, 76015000686028, 283720418696600

Offset

4,3

Links

Alois P. Heinz, Table of n, a(n) for n = 4..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, 4)-g(n, 3):
seq(a(n), n=4..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, 4] - g[n, 3], {n, 4, 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, 4) - g(n, 3)
print([a(n) for n in range(4, 36)]) # Indranil Ghosh, Aug 08 2017

Crossrefs

Column k=4 of A287822.

Cf. A000108.

Sequence in context: A024202 A213775 A133258 * A103738 A348487 A045801

Adjacent sequences: A288742 A288743 A288744 * A288746 A288747 A288748

Keyword

nonn

Author

Alois P. Heinz, Jun 14 2017

Status

approved