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A288749 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals eight. 2
1, 1, 19, 84, 461, 2222, 10577, 48943, 222627, 997735, 4417674, 19359659, 84099436, 362570722, 1552681071, 6609823112, 27989970166, 117967914457, 495087382572, 2069827499508, 8623283249034, 35811917284318, 148289870077879, 612382134256433, 2522591250558641 (list; graph; refs; listen; history; text; internal format)
OFFSET
8,3
LINKS
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, 8)-g(n, 7):
seq(a(n), n=8..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, 8] - g[n, 7], {n, 8, 35}] (* Indranil Ghosh, Aug 08 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, 8) - g(n, 7)
print([a(n) for n in range(8, 36)]) # Indranil Ghosh, Aug 08 2017
CROSSREFS
Column k=8 of A287822.
Cf. A000108.
Sequence in context: A036564 A062639 A209369 * A358932 A039609 A063496
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 14 2017
STATUS
approved

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)