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A288748 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals seven. 2
1, 1, 17, 71, 368, 1697, 7769, 34751, 153313, 668088, 2882104, 12329145, 52358300, 220901081, 926638057, 3867432363, 16068748557, 66495876593, 274178902925, 1126793986670, 4616878543095, 18864740697016, 76885237242318, 312611605360287, 1268261191750753 (list; graph; refs; listen; history; text; internal format)
OFFSET
7,3
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, 7)-g(n, 6):
seq(a(n), n=7..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, 7] - g[n, 6], {n, 7, 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, 7) - g(n, 6)
print([a(n) for n in range(7, 36)]) # Indranil Ghosh, Aug 08 2017
CROSSREFS
Column k=7 of A287822.
Cf. A000108.
Sequence in context: A347334 A290243 A320895 * A069496 A047978 A050524
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 14 2017
STATUS
approved

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Last modified July 17 14:36 EDT 2024. Contains 374377 sequences. (Running on oeis4.)