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Number of Dyck paths of semilength n such that neighboring peaks have different heights.
3

%I #14 Apr 26 2024 18:11:45

%S 1,1,1,3,6,16,43,116,329,947,2762,8176,24469,73896,224960,689623,

%T 2126947,6595273,20548732,64298032,201969823,636639087,2013174695,

%U 6384574754,20302081620,64716553655,206764068578,661983640080,2123573339921,6824597707820,21969711674581

%N Number of Dyck paths of semilength n such that neighboring peaks have different heights.

%H Alois P. Heinz, <a href="/A371705/b371705.txt">Table of n, a(n) for n = 0..555</a>

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

%e a(3) = 3: /\

%e / \ /\ /\

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

%p b:= proc(x, y, h, t) option remember;

%p `if`(y<0 or y>x, 0, `if`(x=0, 1, b(x-1, y+1, h, 1)+

%p `if`(t=0 or y<>h, b(x-1, y-1, `if`(t=1, y, h), 0), 0)))

%p end:

%p a:= n-> b(2*n, 0$3):

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

%Y Cf. A000108, A287776.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Apr 03 2024