A316730 Number of permutations of {0,1,...,2n+2} with first element n whose sequence of ascents and descents forms a Dyck path.

1, 7, 121, 4411, 283073, 28318137, 4076415425, 798519164779, 204292676593353, 66150225395814649, 26444888796754193841, 12792566645739144488693, 7364969554345555373419625, 4976538708651698959601499559, 3900052284443403730374391636689

Offset: 0

Author

Alois P. Heinz, Jul 11 2018

Keywords

• nonn

Comments

All terms are odd.

Examples

			a(0) = 1: 021.
a(1) = 7: 12043, 12430, 13042, 13240, 13420, 14032, 14230.
a(2) = 121: 2301654, 2304165, 2304651, 2305164, ..., 2635041, 2635140, 2645031, 2645130.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..224
• Wikipedia, Counting lattice paths

Crossrefs

Cf. 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, 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+2, 0];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Formula

a(n) = A316728(n+1,n).

a(n) ~ c * 4^n * (n!)^2, where c = 1.897642067924382577976619913635026612792069869805703855808680498665... - Vaclav Kotesovec, Jul 15 2018