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A288325 Number of Dyck paths of semilength n such that each positive level has exactly nine peaks.

%I A288325 #8 Jun 02 2018 10:37:32
%S A288325 1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,10,440,6160,40040,140140,
%T A288325 280280,320320,194480,48620,0,100,9350,382800,9083800,142638320,
%U A288325 1602400800,13556342800,89523519800,473679520600,2047398407340,7334909697400,22016582387800
%N A288325 Number of Dyck paths of semilength n such that each positive level has exactly nine peaks.
%H A288325 Alois P. Heinz, <a href="/A288325/b288325.txt">Table of n, a(n) for n = 0..1000</a>
%H A288325 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288325 b:= proc(n, k, j) option remember;
%p A288325      `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
%p A288325       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288325     end:
%p A288325 a:= n-> `if`(n=0, 1, b(n, 9$2)):
%p A288325 seq(a(n), n=0..42);
%t A288325 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 A288325 a[n_] := If[n == 0, 1, b[n, 9, 9]];
%t A288325 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288325 Column k=9 of A288318.
%Y A288325 Cf. A000108.
%K A288325 nonn
%O A288325 0,20
%A A288325 _Alois P. Heinz_, Jun 07 2017