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A288111 Number of Dyck paths of semilength n such that each level has exactly four peaks or no peaks.

%I A288111 #11 Jun 02 2018 10:36:06
%S A288111 1,0,0,0,1,1,4,7,11,22,81,235,673,2063,5716,13627,33752,95729,298232,
%T A288111 946563,2977953,9147328,27004159,76880498,217826819,637089405,
%U A288111 1949908577,6160707450,19627448025,61909478550,191681762379,583025396879,1756696160636
%N A288111 Number of Dyck paths of semilength n such that each level has exactly four peaks or no peaks.
%H A288111 Alois P. Heinz, <a href="/A288111/b288111.txt">Table of n, a(n) for n = 0..1000</a>
%H A288111 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A288111 . a(6) = 4:
%e A288111 .                                             /\/\/\/\
%e A288111 .  /\  /\/\/\    /\/\  /\/\    /\/\/\  /\    /        \
%e A288111 . /  \/      \  /    \/    \  /      \/  \  /          \  .
%p A288111 b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
%p A288111       b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
%p A288111        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288111     end:
%p A288111 a:= n-> `if`(n=0, 1, b(n, 4$2)):
%p A288111 seq(a(n), n=0..40);
%t A288111 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 A288111 a[n_] := If[n == 0, 1, b[n, 4, 4]];
%t A288111 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288111 Column k=4 of A288108.
%K A288111 nonn
%O A288111 0,7
%A A288111 _Alois P. Heinz_, Jun 05 2017