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%I #8 Jun 02 2018 10:37:15
%S 1,0,0,0,0,0,1,0,0,0,0,0,0,7,140,840,2100,2310,924,0,49,2156,42140,
%T 479220,3598560,19184676,75954564,229873063,541427264,1002386336,
%U 1473318476,1876489398,3785310858,20726607804,136977861097,786065454860,3841493284076
%N Number of Dyck paths of semilength n such that each positive level has exactly six peaks.
%H Alois P. Heinz, <a href="/A288322/b288322.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>
%p b:= proc(n, k, j) option remember;
%p `if`(n=j, 1, add(b(n-j, k, i)*(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, 6$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[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
%t a[n_] := If[n == 0, 1, b[n, 6, 6]];
%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y Column k=6 of A288318.
%Y Cf. A000108.
%K nonn
%O 0,14
%A _Alois P. Heinz_, Jun 07 2017
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