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

%I A288115 #9 Jun 02 2018 10:36:33
%S A288115 1,0,0,0,0,0,0,0,1,1,8,29,71,148,302,596,1101,1915,3485,8991,32879,
%T A288115 131942,595068,2731434,11077722,38438377,117144042,324706536,
%U A288115 842734665,2087025088,4995608093,11799404719,28899101722,79974175125,268545121874,1071998634063
%N A288115 Number of Dyck paths of semilength n such that each level has exactly eight peaks or no peaks.
%H A288115 Alois P. Heinz, <a href="/A288115/b288115.txt">Table of n, a(n) for n = 0..1000</a>
%H A288115 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288115 b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
%p A288115       b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
%p A288115        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288115     end:
%p A288115 a:= n-> `if`(n=0, 1, b(n, 8$2)):
%p A288115 seq(a(n), n=0..37);
%t A288115 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]}]];
%t A288115 a[n_] := If[n == 0, 1, b[n, 8, 8]];
%t A288115 Table[a[n], {n, 0, 37}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288115 Column k=8 of A288108.
%K A288115 nonn
%O A288115 0,11
%A A288115 _Alois P. Heinz_, Jun 05 2017