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A288323 Number of Dyck paths of semilength n such that each positive level has exactly seven peaks.

%I A288323 #8 Jun 02 2018 10:37:20
%S A288323 1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,216,1800,6600,11880,10296,3432,0,64,
%T A288323 3744,96768,1454160,14460480,102586176,544817856,2237725512,
%U A288323 7268659712,18954982080,40057015680,68941928016,97350892224,122456030112,244967552640
%N A288323 Number of Dyck paths of semilength n such that each positive level has exactly seven peaks.
%H A288323 Alois P. Heinz, <a href="/A288323/b288323.txt">Table of n, a(n) for n = 0..1000</a>
%H A288323 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288323 b:= proc(n, k, j) option remember;
%p A288323      `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
%p A288323       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288323     end:
%p A288323 a:= n-> `if`(n=0, 1, b(n, 7$2)):
%p A288323 seq(a(n), n=0..40);
%t A288323 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 A288323 a[n_] := If[n == 0, 1, b[n, 7, 7]];
%t A288323 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288323 Column k=7 of A288318.
%Y A288323 Cf. A000108.
%K A288323 nonn
%O A288323 0,16
%A A288323 _Alois P. Heinz_, Jun 07 2017