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A287993 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and all the level peak numbers are distinct.

%I A287993 #18 May 31 2018 03:08:13
%S A287993 1,1,1,1,6,10,21,52,147,564,1651,4440,12499,36853,116476,390774,
%T A287993 1352215,4593736,15057127,48419013,156073723,511324062,1713185811,
%U A287993 5878350249,20574046540,72771206715,257475113013,905430711156,3160767910928,10981916671027
%N A287993 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and all the level peak numbers are distinct.
%H A287993 Alois P. Heinz, <a href="/A287993/b287993.txt">Table of n, a(n) for n = 0..70</a>
%H A287993 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A287993 a(4) = 6:
%e A287993                      /\      /\      /\          /\/\    /\/\
%e A287993       /\/\/\/\  /\/\/  \  /\/  \/\  /  \/\/\  /\/    \  /    \/\
%p A287993 b:= proc(n, s, j) option remember; `if`(n=j, 1, add(add(
%p A287993        b(n-j, s union {t}, i)*binomial(i, t)*binomial(j-1, i-1-t),
%p A287993        t={$max(1, i-j)..min(n-j, i-1)} minus s), i=1..n-j))
%p A287993     end:
%p A287993 a:= n-> `if`(n=0, 1, add(b(n, {k}, k), k=1..n)):
%p A287993 seq(a(n), n=0..30);
%t A287993 b[n_, s_, j_] := b[n, s, j] = If[n==j, 1, Sum[Sum[b[n-j, s ~Union~ {t}, i]* Binomial[i, t]*Binomial[j-1, i-1-t], {t, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ s}], {i, 1, n - j}]];
%t A287993 a[n_] := If[n == 0, 1, Sum[b[n, {k}, k], {k, 1, n}]];
%t A287993 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A287993 Cf. A000108, A287987, A287989.
%K A287993 nonn
%O A287993 0,5
%A A287993 _Alois P. Heinz_, Jun 04 2017