A288748 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals seven.

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

Offset

7,3

Links

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Wikipedia, Counting lattice paths

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

Adjacent sequences: A288745 A288746 A288747 * A288749 A288750 A288751

Keyword

nonn

Author

Alois P. Heinz, Jun 14 2017

Status

approved