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%I #31 Mar 04 2024 15:00:33
%S 1,1,0,2,0,4,6,8,24,52,96,212,504,1072,2352,5288,11928,26800,60336,
%T 136304,308928,701248,1593120,3622016,8245008,18787360,42836928,
%U 97724384,223052784,509338816,1163512032,2658731648,6077117376,13893874624,31771515648
%N Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly one peak.
%C All terms with n > 1 are even.
%H Alois P. Heinz, <a href="/A287846/b287846.txt">Table of n, a(n) for n = 0..1000</a>
%H Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 15.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e . a(1) = 1: /\ .
%e .
%e . a(3) = 2: /\ /\
%e . /\/ \ / \/\ .
%e .
%e . a(5) = 4:
%e . /\ /\ /\ /\
%e . /\/ \ / \/\ /\/ \ / \/\
%e . /\/ \ /\/ \ / \/\ / \/\ .
%p b:= proc(n, j) option remember; `if`(n=j or n=0, 1, add(
%p b(n-j, i)*binomial(j-1, i-2)*i, i=1..min(j+2, n-j)))
%p end:
%p a:= n-> b(n, 1):
%p seq(a(n), n=0..35);
%t b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*Binomial[j - 1, i - 2]*i, {i, 1, Min[j + 2, n - j]}]];
%t a[n_] := b[n, 1];
%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 23 2018, translated from Maple *)
%Y Column k=1 of A288318.
%Y Cf. A000108, A281874, A287843, A287845, A287901, A287963, A287987, A289020.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Jun 01 2017
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