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A288744
Number of Dyck paths of semilength n such that the maximal number of peaks per level equals three.
2
1, 1, 9, 29, 112, 405, 1514, 5565, 20249, 73416, 265616, 957677, 3441282, 12329838, 44062706, 157105923, 559009643, 1985301783, 7038496811, 24913917722, 88058727525, 310832221932, 1095854282575, 3859201682187, 13576884290502, 47719628447310, 167579774234059
OFFSET
3,3
LINKS
EXAMPLE
. T(4) = 1: /\/\/\
. / \ .
MAPLE
b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 3)-g(n, 2):
seq(a(n), n=3..35);
MATHEMATICA
b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 3] - g[n, 2], {n, 3, 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(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
def a(n): return g(n, 3) - g(n, 2)
print([a(n) for n in range(3, 36)]) # Indranil Ghosh, Aug 09 2017
CROSSREFS
Column k=3 of A287822.
Cf. A000108.
Sequence in context: A147268 A147376 A201447 * A183245 A101141 A056258
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 14 2017
STATUS
approved