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A287963 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has one or two peaks.

%I A287963 #10 May 29 2018 08:28:36
%S A287963 1,1,1,2,5,10,28,71,194,532,1495,4256,12176,35251,102664,300260,
%T A287963 881909,2599948,7688164,22788527,67676144,201308938,599676445,
%U A287963 1788564038,5339905904,15956230705,47713265536,142763240666,427390085963,1280058256294,3835332884686
%N A287963 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has one or two peaks.
%H A287963 Alois P. Heinz, <a href="/A287963/b287963.txt">Table of n, a(n) for n = 0..1000</a>
%H A287963 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A287963 . a(3) = 2:     /\      /\
%e A287963 .            /\/  \    /  \/\  .
%e A287963 .
%e A287963 . a(4) = 5:      /\      /\        /\/\    /\        /\/\
%e A287963 .           /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /    \/\ .
%p A287963 b:= proc(n, j) option remember; `if`(n=j, 1, add(
%p A287963        b(n-j, i)*i*(binomial(j-1, i-2) +(i-1)/2*
%p A287963        binomial(j-1, i-3)), i=2..min(j+3, n-j)))
%p A287963     end:
%p A287963 a:= n-> `if`(n=0, 1, b(n, 1)+b(n, 2)):
%p A287963 seq(a(n), n=0..35);
%t A287963 b[n_, j_] := b[n, j] = If[n == j, 1, Sum[b[n - j, i]*i*(Binomial[j - 1, i - 2] + (i - 1)/2*Binomial[j - 1, i - 3]), {i, 2, Min[j + 3, n - j]}]];
%t A287963 a[n_] := If[n == 0, 1, b[n, 1] + b[n, 2]];
%t A287963 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 29 2018, from Maple *)
%Y A287963 Cf. A000108, A281874, A287843, A287845, A287846.
%K A287963 nonn
%O A287963 0,4
%A A287963 _Alois P. Heinz_, Jun 03 2017