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A288320 Number of Dyck paths of semilength n such that each positive level has exactly four peaks.

%I A288320 #9 Jun 02 2018 10:37:02
%S A288320 1,0,0,0,1,0,0,0,0,5,45,105,70,0,25,525,4950,26950,94605,226925,
%T A288320 383525,507000,1016475,5047875,26940475,117108550,414703200,
%U A288320 1223146475,3089625550,7073320775,16715232600,48599763900,175648700900,673443954000,2444611549450
%N A288320 Number of Dyck paths of semilength n such that each positive level has exactly four peaks.
%H A288320 Alois P. Heinz, <a href="/A288320/b288320.txt">Table of n, a(n) for n = 0..1000</a>
%H A288320 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288320 b:= proc(n, k, j) option remember;
%p A288320      `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
%p A288320       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288320     end:
%p A288320 a:= n-> `if`(n=0, 1, b(n, 4$2)):
%p A288320 seq(a(n), n=0..35);
%t A288320 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 A288320 a[n_] := If[n == 0, 1, b[n, 4, 4]];
%t A288320 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288320 Column k=4 of A288318.
%Y A288320 Cf. A000108.
%K A288320 nonn
%O A288320 0,10
%A A288320 _Alois P. Heinz_, Jun 07 2017