A316727 Number of permutations of {0,1,...,2n} with first element n whose sequence of ascents and descents forms a Dyck path.

1, 1, 5, 87, 3337, 223333, 23068057, 3403720071, 679894572497, 176710079709345, 57967294285022281, 23427042148948682599, 11437832700333350250001, 6637473822604173137681381, 4515971399162518697397538173, 3560540787622773257563653593551

Offset: 0

Author

Alois P. Heinz, Jul 11 2018

Keywords

• nonn

Comments

All terms are odd.

Examples

			a(0) = 1: 0.
a(1) = 1: 120.
a(2) = 5: 23041, 23140, 23410, 24031, 24130.
a(3) = 87: 3401652, 3402165, 3402651, 3405162, ..., 3625041, 3625140, 3645021, 3645120.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..225
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A303285, A316728.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
    If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
    If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, n, 0];
a /@ Range[0, 20] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

Formula

a(n) = A316728(n,n).

a(n) ~ c * 4^n * (n!)^2 / n^2, where c = 0.47441051698109564449415497840875665319801746745142596395217012466627... - Vaclav Kotesovec, Jul 15 2018