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A288112 Number of Dyck paths of semilength n such that each level has exactly five peaks or no peaks.

%I A288112 #8 Jun 02 2018 10:36:12
%S A288112 1,0,0,0,0,1,1,5,11,19,32,60,178,612,1910,6505,22097,62717,155341,
%T A288112 365413,908850,2587326,8337462,28613490,99865122,341887279,1112148217,
%U A288112 3385839203,9723179369,27116765041,76656520298,228493968174,728697760582,2447359432110
%N A288112 Number of Dyck paths of semilength n such that each level has exactly five peaks or no peaks.
%H A288112 Alois P. Heinz, <a href="/A288112/b288112.txt">Table of n, a(n) for n = 0..1000</a>
%H A288112 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288112 b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
%p A288112       b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
%p A288112        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288112     end:
%p A288112 a:= n-> `if`(n=0, 1, b(n, 5$2)):
%p A288112 seq(a(n), n=0..37);
%t A288112 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 A288112 a[n_] := If[n == 0, 1, b[n, 5, 5]];
%t A288112 Table[a[n], {n, 0, 37}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288112 Column k=5 of A288108.
%K A288112 nonn
%O A288112 0,8
%A A288112 _Alois P. Heinz_, Jun 05 2017