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A288682
Number of Dyck paths of semilength n such that no positive level has fewer than six peaks.
2
1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 8, 156, 1213, 5232, 16091, 41834, 100320, 229851, 513699, 1166304, 3068322, 11294356, 54431307, 271824026, 1253186445, 5233138157, 20031588131, 71538367677, 242280234545, 789260222205, 2507719402158, 7900354628357
OFFSET
0,14
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, 6, j], {j, 6, 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, 6, j) for j in range(6, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 10 2017
CROSSREFS
Column k=6 of A288386.
Cf. A000108.
Sequence in context: A302959 A188408 A089669 * A268543 A345317 A113668
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 13 2017
STATUS
approved