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A288109 Number of Dyck paths of semilength n such that all levels with peaks have exactly the same number of peaks.

%I A288109 #10 May 31 2018 03:08:19
%S A288109 1,1,2,5,9,23,56,122,323,792,2060,5199,13314,35171,94077,249285,
%T A288109 662901,1775244,4806724,13125887,36107283,99863241,276784435,
%U A288109 768288783,2143763275,6037486060,17171063218,49187617277,141512589597,408293870713,1181084207303
%N A288109 Number of Dyck paths of semilength n such that all levels with peaks have exactly the same number of peaks.
%H A288109 Alois P. Heinz, <a href="/A288109/b288109.txt">Table of n, a(n) for n = 0..300</a>
%H A288109 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A288109 b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
%p A288109       b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
%p A288109        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
%p A288109     end:
%p A288109 a:= n-> 1 + add(b(n, j$2), j=1..n-1):
%p A288109 seq(a(n), n=0..33);
%t A288109 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 A288109 a[n_] := 1 + Sum[b[n, j, j], {j, 1, n - 1}];
%t A288109 Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A288109 Row sums of A288108.
%Y A288109 Cf. A000108, A287987.
%K A288109 nonn
%O A288109 0,3
%A A288109 _Alois P. Heinz_, Jun 05 2017