A288140 Number of Dyck paths of semilength n such that the number of peaks is weakly decreasing from lower to higher levels.

1, 1, 1, 3, 4, 12, 28, 63, 177, 455, 1233, 3383, 9359, 26809, 77078, 223201, 653982, 1934508, 5783712, 17431660, 52879184, 161386859, 495432345, 1530191918, 4754079840, 14849407892, 46604383972, 146897291083, 464892421363, 1477052536749, 4711124635655

Offset: 0

Author

Alois P. Heinz, Jun 05 2017

Keywords

• nonn

Examples

			. a(5) = 12:
.                       /\         /\         /\
.   /\/\/\/\/\   /\/\/\/  \   /\/\/  \/\   /\/  \/\/\
.
.    /\               /\/\       /\/\       /\/\
.   /  \/\/\/\   /\/\/    \   /\/    \/\   /    \/\/\
.
.         /\         /\           /\         /\
.      /\/  \       /  \/\     /\/  \       /  \/\
.   /\/      \   /\/      \   /      \/\   /      \/\  .
		

Links

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

Crossrefs

Cf. A000108, A008930, A048285, A288141, A288146, 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(k, i-j)..min(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..31);

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[k, i - j], Min[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, 31}] (* Jean-François Alcover, May 29 2018, from Maple *)