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A288117 Number of Dyck paths of semilength n such that each level has exactly ten peaks or no peaks.

%I A288117 #9 Jun 02 2018 10:36:46
%S A288117 1,0,0,0,0,0,0,0,0,0,1,1,10,46,139,337,760,1672,3511,6904,12782,22627,
%T A288117 39318,75538,207881,812507,3841220,20378041,105202200,477568717,
%U A288117 1875871984,6503542301,20447457784,59755535440,165351799936,437965172476,1115247676801
%N A288117 Number of Dyck paths of semilength n such that each level has exactly ten peaks or no peaks.
%H A288117 Alois P. Heinz, <a href="/A288117/b288117.txt">Table of n, a(n) for n = 0..1000</a>
%H A288117 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288117 b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
%p A288117       b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
%p A288117        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288117     end:
%p A288117 a:= n-> `if`(n=0, 1, b(n, 10$2)):
%p A288117 seq(a(n), n=0..40);
%t A288117 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 A288117 a[n_] := If[n == 0, 1, b[n, 10, 10]];
%t A288117 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A288117 Column k=10 of A288108.
%K A288117 nonn
%O A288117 0,13
%A A288117 _Alois P. Heinz_, Jun 05 2017