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A287776 Number of Dyck paths of semilength n such that every peak at level y > 1 is preceded by (at least) one peak at level y-1 and is followed by (at least) one peak at level y-1.

%I A287776 #19 Mar 29 2024 10:44:57
%S A287776 1,1,1,1,2,4,8,17,39,95,241,629,1679,4572,12684,35812,102774,299371,
%T A287776 883848,2641121,7978262,24337821,74908008,232451921,726831776,
%U A287776 2288799963,7255401745,23143158678,74256591422,239582207959,777047305709,2532730030266,8293970682858
%N A287776 Number of Dyck paths of semilength n such that every peak at level y > 1 is preceded by (at least) one peak at level y-1 and is followed by (at least) one peak at level y-1.
%H A287776 Alois P. Heinz, <a href="/A287776/b287776.txt">Table of n, a(n) for n = 0..430</a>
%H A287776 Axel Bacher, <a href="https://arxiv.org/abs/2403.08120">Progressive and rushed Dyck paths</a>, arXiv:2403.08120 [math.CO], 2024. See p. 7.
%H A287776 Alois P. Heinz, <a href="/A287776/a287776.gif">Animation of a(9) = 95 paths</a>
%H A287776 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A287776  a(4) = 2:                 /\
%e A287776             /\/\/\/\    /\/  \/\  .
%e A287776  a(5) = 4:                   /\        /\          /\/\
%e A287776             /\/\/\/\/\  /\/\/  \/\  /\/  \/\/\  /\/    \/\  .
%p A287776 b:= proc(x, y, k, h, t) option remember; `if`(x=0, `if`(h<2, 1, 0),
%p A287776       `if`(y<=k and y<x-1, b(x-1, y+1, k, h, 1), 0) +`if`(y>0,
%p A287776        b(x-1, y-1, max(y, k), `if`(t=1 and h<=y+1, y, h), 0), 0))
%p A287776     end:
%p A287776 a:= n-> b(2*n, 0$4):
%p A287776 seq(a(n), n=0..35);
%t A287776 b[x_, y_, k_, h_, t_] := b[x, y, k, h, t] = If[x == 0, If[h < 2, 1, 0], If[y <= k && y < x - 1, b[x - 1, y + 1, k, h, 1], 0] + If[y > 0, b[x - 1, y - 1, Max[y, k], If[t == 1 && h <= y + 1, y, h], 0], 0]];
%t A287776 a[n_] := b[2n, 0, 0, 0, 0];
%t A287776 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%Y A287776 Cf. A000108, A287709.
%K A287776 nonn
%O A287776 0,5
%A A287776 _Alois P. Heinz_, May 31 2017