A288141 Number of Dyck paths of semilength n such that the number of peaks is strongly decreasing from lower to higher levels.

1, 1, 1, 1, 4, 5, 10, 22, 46, 148, 324, 722, 1843, 4634, 12537, 34248, 95711, 266761, 724689, 1983267, 5553902, 15900083, 46201546, 135511171, 400668869, 1189723253, 3535186203, 10516298421, 31405658622, 94378367065, 285623516777, 870481565252, 2671088133010

Offset

0,5

Links

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

Wikipedia, Counting lattice paths

Example

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

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+1, 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..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[k + 1, 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, 34}] (* Jean-François Alcover, May 29 2018, from Maple *)

Crossrefs

Cf. A000108, A008930, A048285, A288140, A288146, A288147.

Sequence in context: A049898 A166577 A242960 * A203853 A109675 A052508

Adjacent sequences: A288138 A288139 A288140 * A288142 A288143 A288144

Keyword

nonn

Author

Alois P. Heinz, Jun 05 2017

Status

approved