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A288319 Number of Dyck paths of semilength n such that each positive level has exactly three peaks.

%I A288319 #9 Jun 02 2018 10:36:52
%S A288319 1,0,0,1,0,0,0,4,20,20,0,16,200,1120,3540,6864,9400,18240,82000,
%T A288319 364256,1255040,3448400,8094400,18653984,50789120,166596240,565558400,
%U A288319 1791310496,5202559520,14279014880,39040502400,111437733184,335085082880,1032287357600
%N A288319 Number of Dyck paths of semilength n such that each positive level has exactly three peaks.
%H A288319 Alois P. Heinz, <a href="/A288319/b288319.txt">Table of n, a(n) for n = 0..1000</a>
%H A288319 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A288319 . a(7) = 4:
%e A288319 .           /\/\/\        /\/\/\        /\/\/\        /\/\/\
%e A288319 .    /\/\/\/      \  /\/\/      \/\  /\/      \/\/\  /      \/\/\/\ .
%p A288319 b:= proc(n, k, j) option remember;
%p A288319      `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
%p A288319       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288319     end:
%p A288319 a:= n-> `if`(n=0, 1, b(n, 3$2)):
%p A288319 seq(a(n), n=0..35);
%t A288319 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 A288319 a[n_] := If[n == 0, 1, b[n, 3, 3]];
%t A288319 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288319 Column k=3 of A288318.
%Y A288319 Cf. A000108.
%K A288319 nonn
%O A288319 0,8
%A A288319 _Alois P. Heinz_, Jun 07 2017