login
A288679
Number of Dyck paths of semilength n such that no positive level has fewer than three peaks.
2
1, 0, 0, 1, 1, 1, 1, 5, 30, 96, 245, 592, 1543, 4884, 17660, 64495, 226442, 766937, 2558655, 8590293, 29408344, 102893203, 366035420, 1314955687, 4747101946, 17184305311, 62359953380, 226978626707, 829122987011, 3040369502702, 11191473790567, 41342469523031
OFFSET
0,8
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, 3, j], {j, 3, 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, 3, j) for j in range(3, n + 1)])
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 09 2017
CROSSREFS
Column k=3 of A288386.
Cf. A000108.
Sequence in context: A164015 A128302 A258582 * A071252 A174002 A030506
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 13 2017
STATUS
approved