A303286 Number of permutations p of [2n+1] such that the sequence of ascents and descents of 0p0 forms a Dyck path.

1, 4, 60, 1974, 114972, 10490392, 1384890104, 250150900354, 59317740001132, 17886770092245360, 6687689652133397064, 3037468107154650475868, 1647659575564603380270360, 1052309674407466474533397824, 781725844087366504901991503920, 668408235613132734111402947167658

Offset: 0

Author

Alois P. Heinz, Apr 20 2018

Keywords

• nonn

Examples

			a(1) = 4: 132, 213, 231, 312.
		

Links

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

Crossrefs

Bisection (odd part) of A303284.

Cf. A180056.

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(0, 2*n+1, 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[0, 2*n + 1, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 31 2018, from Maple *)

Formula

a(n) ~ c * 2^(2*n) * n^(2*n) / exp(2*n), where c = 45.0971960423271758887353825240016439879529954831112316... - Vaclav Kotesovec, May 22 2018