A288680 Number of Dyck paths of semilength n such that no positive level has fewer than four peaks.

1, 0, 0, 0, 1, 1, 1, 1, 1, 6, 57, 247, 718, 1795, 4210, 9969, 27596, 98507, 402924, 1626525, 6142611, 21729644, 73308577, 241270869, 793679894, 2666563900, 9263663359, 33259282181, 122178034000, 453573262015, 1685632454779, 6240174176549, 22987207140830

Offset

0,10

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Wikipedia, Counting lattice paths

Mathematica

b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[k, i - j], i - 1}] b[n - j, k, i], {i, n - j}]];  a[n_]:=If[n==0, 1, Sum[b[n, 4, j], {j, 4, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Aug 09 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([sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(k, i - j), i)])*b(n - j, k, i) for i in range(1, n - j + 1)])
def a(n): return 1 if n==0 else sum([b(n, 4, j) for j in range(4, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 09 2017

Crossrefs

Column k=4 of A288386.

Cf. A000108.

Sequence in context: A137032 A053421 A083696 * A181430 A281557 A227813

Adjacent sequences: A288677 A288678 A288679 * A288681 A288682 A288683

Keyword

nonn

Author

Alois P. Heinz, Jun 13 2017

Status

approved