A288115 Number of Dyck paths of semilength n such that each level has exactly eight peaks or no peaks.

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 8, 29, 71, 148, 302, 596, 1101, 1915, 3485, 8991, 32879, 131942, 595068, 2731434, 11077722, 38438377, 117144042, 324706536, 842734665, 2087025088, 4995608093, 11799404719, 28899101722, 79974175125, 268545121874, 1071998634063

Offset

0,11

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Counting lattice paths

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
      b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
    end:
a:= n-> `if`(n=0, 1, b(n, 8$2)):
seq(a(n), n=0..37);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
a[n_] := If[n == 0, 1, b[n, 8, 8]];
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Column k=8 of A288108.

Sequence in context: A093809 A244244 A037157 * A100178 A106113 A362153

Adjacent sequences: A288112 A288113 A288114 * A288116 A288117 A288118

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved