A287993 - OEIS

%I #18 May 31 2018 03:08:13

%S 1,1,1,1,6,10,21,52,147,564,1651,4440,12499,36853,116476,390774,

%T 1352215,4593736,15057127,48419013,156073723,511324062,1713185811,

%U 5878350249,20574046540,72771206715,257475113013,905430711156,3160767910928,10981916671027

%N 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 Alois P. Heinz, <a href="/A287993/b287993.txt">Table of n, a(n) for n = 0..70</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%e a(4) = 6:

%e                      /\      /\      /\          /\/\    /\/\

%e       /\/\/\/\  /\/\/  \  /\/  \/\  /  \/\/\  /\/    \  /    \/\

%p b:= proc(n, s, j) option remember; `if`(n=j, 1, add(add(

%p        b(n-j, s union {t}, i)*binomial(i, t)*binomial(j-1, i-1-t),

%p        t={$max(1, i-j)..min(n-j, i-1)} minus s), i=1..n-j))

%p     end:

%p a:= n-> `if`(n=0, 1, add(b(n, {k}, k), k=1..n)):

%p seq(a(n), n=0..30);

%t 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 a[n_] := If[n == 0, 1, Sum[b[n, {k}, k], {k, 1, n}]];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 31 2018, from Maple *)

%Y Cf. A000108, A287987, A287989.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Jun 04 2017