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A371705 Number of Dyck paths of semilength n such that neighboring peaks have different heights.

%I A371705 #14 Apr 26 2024 18:11:45
%S A371705 1,1,1,3,6,16,43,116,329,947,2762,8176,24469,73896,224960,689623,
%T A371705 2126947,6595273,20548732,64298032,201969823,636639087,2013174695,
%U A371705 6384574754,20302081620,64716553655,206764068578,661983640080,2123573339921,6824597707820,21969711674581
%N A371705 Number of Dyck paths of semilength n such that neighboring peaks have different heights.
%H A371705 Alois P. Heinz, <a href="/A371705/b371705.txt">Table of n, a(n) for n = 0..555</a>
%H A371705 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A371705 a(3) = 3:    /\
%e A371705             /  \        /\            /\
%e A371705            /    \      /  \/\      /\/  \    .
%p A371705 b:= proc(x, y, h, t) option remember;
%p A371705      `if`(y<0 or y>x, 0, `if`(x=0, 1, b(x-1, y+1, h, 1)+
%p A371705      `if`(t=0 or y<>h, b(x-1, y-1, `if`(t=1, y, h), 0), 0)))
%p A371705     end:
%p A371705 a:= n-> b(2*n, 0$3):
%p A371705 seq(a(n), n=0..30);
%Y A371705 Cf. A000108, A287776.
%K A371705 nonn
%O A371705 0,4
%A A371705 _Alois P. Heinz_, Apr 03 2024