A288146 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.

1, 1, 1, 3, 3, 13, 28, 65, 199, 540, 1468, 4188, 12328, 36870, 110181, 331226, 1012241, 3137822, 9796834, 30695164, 96658857, 306575170, 979485119, 3148413910, 10169223709, 32983822120, 107413795300, 351235602807, 1153308804255, 3802294411213, 12581993628872

Offset: 0

Author

Alois P. Heinz, Jun 05 2017

Keywords

• nonn

Examples

			a(3) = 3:
                           /\       /\
             /\/\/\     /\/  \     /  \/\
a(4) = 3:
                          /\/\     /\/\
            /\/\/\/\   /\/    \   /    \/\
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Wikipedia, Counting lattice paths

Crossrefs

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

Programs

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       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))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..34);

Mathematica

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}]];
a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 29 2018, from Maple *)