A287993 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and all the level peak numbers are distinct.

1, 1, 1, 1, 6, 10, 21, 52, 147, 564, 1651, 4440, 12499, 36853, 116476, 390774, 1352215, 4593736, 15057127, 48419013, 156073723, 511324062, 1713185811, 5878350249, 20574046540, 72771206715, 257475113013, 905430711156, 3160767910928, 10981916671027

Offset: 0

Author

Alois P. Heinz, Jun 04 2017

Keywords

• nonn

Examples

			a(4) = 6:
                     /\      /\      /\          /\/\    /\/\
      /\/\/\/\  /\/\/  \  /\/  \/\  /  \/\/\  /\/    \  /    \/\
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..70
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A000108, A287987, A287989.

Programs

Maple

b:= proc(n, s, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, s union {t}, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t={$max(1, i-j)..min(n-j, i-1)} minus s), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, {k}, k), k=1..n)):
seq(a(n), n=0..30);

Mathematica

b[n_, s_, j_] := b[n, s, j] = If[n==j, 1, Sum[Sum[b[n-j, s ~Union~ {t}, i]* Binomial[i, t]*Binomial[j-1, i-1-t], {t, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ s}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, {k}, k], {k, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 31 2018, from Maple *)