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A288324 Number of Dyck paths of semilength n such that each positive level has exactly eight peaks.

%I A288324 #8 Jun 02 2018 10:37:26
%S A288324 1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,315,3465,17325,45045,63063,45045,
%T A288324 12870,0,81,6075,200340,3835755,48617415,440531784,3000152925,
%U A288324 15896972520,67174514550,230430986514,649879542063,1519950287430,2963421671535,4828750295985
%N A288324 Number of Dyck paths of semilength n such that each positive level has exactly eight peaks.
%H A288324 Alois P. Heinz, <a href="/A288324/b288324.txt">Table of n, a(n) for n = 0..1000</a>
%H A288324 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288324 b:= proc(n, k, j) option remember;
%p A288324      `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
%p A288324       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288324     end:
%p A288324 a:= n-> `if`(n=0, 1, b(n, 8$2)):
%p A288324 seq(a(n), n=0..40);
%t A288324 b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
%t A288324 a[n_] := If[n == 0, 1, b[n, 8, 8]];
%t A288324 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288324 Column k=8 of A288318.
%Y A288324 Cf. A000108.
%K A288324 nonn
%O A288324 0,18
%A A288324 _Alois P. Heinz_, Jun 07 2017