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A371726 Number of Dyck paths of semilength n such that neighboring peaks differ in height by at most one and first and last peak are at height one.

%I A371726 #21 Apr 25 2024 09:11:21
%S A371726 1,1,1,1,2,4,8,17,39,93,227,567,1447,3760,9917,26492,71567,195256,
%T A371726 537397,1490614,4163508,11702411,33078992,93986117,268294604,
%U A371726 769171970,2213834911,6395017291,18534987580,53887644375,157121564325,459351495858,1346290327899
%N A371726 Number of Dyck paths of semilength n such that neighboring peaks differ in height by at most one and first and last peak are at height one.
%H A371726 Alois P. Heinz, <a href="/A371726/b371726.txt">Table of n, a(n) for n = 0..2031</a>
%H A371726 Alois P. Heinz, <a href="/A371726/a371726.gif">Animation of a(9)=93 paths</a>
%H A371726 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A371726 a(4) = 2:                     /\
%e A371726              /\/\/\/\      /\/  \/\      .
%p A371726 b:= proc(x, y, h, t) option remember; `if`(y<0 or y>x, 0,
%p A371726       `if`(x=0, `if`(h>1, 0, 1), `if`(t=1 and abs(y-h)>1, 0,
%p A371726          b(x-1, y-1, `if`(t=1, y, h), 0))+b(x-1, y+1, h, 1)))
%p A371726     end:
%p A371726 a:= n-> b(2*n, 0$3):
%p A371726 seq(a(n), n=0..32);
%Y A371726 Cf. A000108, A287776, A371705, A371727.
%K A371726 nonn
%O A371726 0,5
%A A371726 _Alois P. Heinz_, Apr 04 2024