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%I #10 Jun 02 2018 10:36:00
%S 1,0,0,1,1,3,4,10,36,83,225,573,1444,3996,11840,34057,95573,267643,
%T 754744,2167250,6347944,18754719,55183269,161366349,471263668,
%U 1382569548,4085677052,12145287569,36193473369,107824201547,320874528844,954819540526,2845349212512
%N Number of Dyck paths of semilength n such that each level has exactly three peaks or no peaks.
%H Alois P. Heinz, <a href="/A288110/b288110.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e a(5) = 3:
%e . /\/\/\
%e . /\ /\/\ /\/\ /\ / \
%e . / \/ \ / \/ \ / \ .
%p b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
%p b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
%p *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p end:
%p a:= n-> `if`(n=0, 1, b(n, 3$2)):
%p seq(a(n), n=0..40);
%t 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 a[n_] := If[n == 0, 1, b[n, 3, 3]];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y Column k=3 of A288108.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Jun 05 2017
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