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A371926 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.

%I A371926 #16 Apr 26 2024 18:11:58
%S A371926 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 A371926 3084,5273,9060,15658,27152,47122,81769,141919,246525,428742,746479,
%U A371926 1300806,2268169,3956840,6905817,12057999,21063319,36809385,64350631,112535774
%N A371926 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 A371926 Alois P. Heinz, <a href="/A371926/b371926.txt">Table of n, a(n) for n = 0..2500</a>
%H A371926 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A371926 a(7) = 1:           /\
%e A371926                  /\/  \/\
%e A371926               /\/        \/\
%e A371926 a(8) = 2:           /\                      /\
%e A371926                  /\/  \  /\            /\  /  \/\
%e A371926               /\/      \/  \/\      /\/  \/      \/\
%e A371926 a(9) = 1:           /\
%e A371926                /\  /  \  /\
%e A371926             /\/  \/    \/  \/\
%e A371926 a(10) = 1:          /\
%e A371926                  /\/  \/\
%e A371926               /\/        \/\
%e A371926            /\/              \/\     .
%p A371926 b:= proc(x, y, v) option remember; (t-> `if`(x=t, 1,
%p A371926      `if`(x<t, 0, add(b(x-1-2*i, y-1, 0), i=1..y-1)+
%p A371926      `if`(v=1, add(b(x-1-2*i, y+1, v), i=1..y), 0))))(3*y-2)
%p A371926     end:
%p A371926 a:= n-> `if`(n=0, 1, b(2*n-1, 1$2)):
%p A371926 seq(a(n), n=0..50);
%Y A371926 Cf. A000108, A371727.
%K A371926 nonn
%O A371926 0,9
%A A371926 _Alois P. Heinz_, Apr 12 2024