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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Wikipedia, Counting lattice paths

Example

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

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 *)

Crossrefs

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

Sequence in context: A217246 A186743 A219314 * A019154 A255675 A272857

Adjacent sequences: A288143 A288144 A288145 * A288147 A288148 A288149

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved