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A288681
Number of Dyck paths of semilength n such that no positive level has fewer than five peaks.
2
1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 7, 98, 575, 2009, 5468, 13365, 30910, 70156, 170830, 531334, 2203895, 10091063, 44034478, 176213307, 650957418, 2258314543, 7491190627, 24204620623, 77794583961, 254583038843, 865776314524, 3087754003802, 11479621448305
OFFSET
0,12
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, 5, j], {j, 5, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Aug 10 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, 5, j) for j in range(5, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 10 2017
CROSSREFS
Column k=5 of A288386.
Cf. A000108.
Sequence in context: A200504 A267641 A267669 * A124092 A036293 A133679
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 13 2017
STATUS
approved