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Number of Dyck paths of semilength n with strongly unimodal peak heights such that neighboring peaks differ in height by exactly one and first and last peak are at height one.
2

%I #16 Apr 26 2024 18:11:58

%S 1,1,0,0,1,0,0,1,2,1,1,4,8,11,14,23,44,79,130,209,347,598,1042,1801,

%T 3084,5273,9060,15658,27152,47122,81769,141919,246525,428742,746479,

%U 1300806,2268169,3956840,6905817,12057999,21063319,36809385,64350631,112535774

%N Number of Dyck paths of semilength n with strongly unimodal peak heights such that neighboring peaks differ in height by exactly one and first and last peak are at height one.

%H Alois P. Heinz, <a href="/A371926/b371926.txt">Table of n, a(n) for n = 0..2500</a>

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

%e a(7) = 1: /\

%e /\/ \/\

%e /\/ \/\

%e a(8) = 2: /\ /\

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

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

%e a(9) = 1: /\

%e /\ / \ /\

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

%e a(10) = 1: /\

%e /\/ \/\

%e /\/ \/\

%e /\/ \/\ .

%p b:= proc(x, y, v) option remember; (t-> `if`(x=t, 1,

%p `if`(x<t, 0, add(b(x-1-2*i, y-1, 0), i=1..y-1)+

%p `if`(v=1, add(b(x-1-2*i, y+1, v), i=1..y), 0))))(3*y-2)

%p end:

%p a:= n-> `if`(n=0, 1, b(2*n-1, 1$2)):

%p seq(a(n), n=0..50);

%Y Cf. A000108, A371727.

%K nonn

%O 0,9

%A _Alois P. Heinz_, Apr 12 2024