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A288680
Number of Dyck paths of semilength n such that no positive level has fewer than four peaks.
2
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 13 2017
STATUS
approved