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Number of Dyck paths of semilength n such that the number of peaks is weakly increasing from lower to higher levels and no positive level is peakless.
4

%I #21 May 29 2018 03:49:40

%S 1,1,1,3,3,13,28,65,199,540,1468,4188,12328,36870,110181,331226,

%T 1012241,3137822,9796834,30695164,96658857,306575170,979485119,

%U 3148413910,10169223709,32983822120,107413795300,351235602807,1153308804255,3802294411213,12581993628872

%N Number of Dyck paths of semilength n such that the number of peaks is weakly increasing from lower to higher levels and no positive level is peakless.

%H Alois P. Heinz, <a href="/A288146/b288146.txt">Table of n, a(n) for n = 0..200</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%e a(3) = 3:

%e /\ /\

%e /\/\/\ /\/ \ / \/\

%e a(4) = 3:

%e /\/\ /\/\

%e /\/\/\/\ /\/ \ / \/\

%p b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(

%p b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),

%p t=max(1, i-j)..min(k, n-j, i-1)), i=1..n-j))

%p end:

%p a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):

%p seq(a(n), n=0..34);

%t b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Max[1, i - j], Min[k, n - j, i - 1]}], {i, 1, n - j}]];

%t a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];

%t Table[a[n], {n, 0, 34}] (* _Jean-François Alcover_, May 29 2018, from Maple *)

%Y Cf. A000108, A008930, A048285, A288140, A288141, A288147.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jun 05 2017