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%I #9 Jun 02 2018 10:36:39
%S 1,0,0,0,0,0,0,0,0,1,1,9,37,101,227,487,1019,2015,3724,6528,11438,
%T 24758,81106,330810,1542486,7723906,35765450,142808117,494994177,
%U 1533142713,4370885515,11737660709,30111369545,74286138919,177289070957,416431652499
%N Number of Dyck paths of semilength n such that each level has exactly nine peaks or no peaks.
%H Alois P. Heinz, <a href="/A288116/b288116.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; `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, 9$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, 9, 9]];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y Column k=9 of A288108.
%K nonn
%O 0,12
%A _Alois P. Heinz_, Jun 05 2017
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