%I #16 Jan 03 2021 06:39:52

%S 1,1,5,87,3337,223333,23068057,3403720071,679894572497,

%T 176710079709345,57967294285022281,23427042148948682599,

%U 11437832700333350250001,6637473822604173137681381,4515971399162518697397538173,3560540787622773257563653593551

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

%C All terms are odd.

%H Alois P. Heinz, <a href="/A316727/b316727.txt">Table of n, a(n) for n = 0..225</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

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

%F a(n) ~ c * 4^n * (n!)^2 / n^2, where c = 0.47441051698109564449415497840875665319801746745142596395217012466627... - _Vaclav Kotesovec_, Jul 15 2018

%e a(0) = 1: 0.

%e a(1) = 1: 120.

%e a(2) = 5: 23041, 23140, 23410, 24031, 24130.

%e a(3) = 87: 3401652, 3402165, 3402651, 3405162, ..., 3625041, 3625140, 3645021, 3645120.

%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,

%p `if`(t>0, add(b(u-j, o+j-1, t-1), j=1..u), 0)+

%p `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..20);

%t b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,

%t If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +

%t If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];

%t a[n_] := b[n, n, 0];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)

%Y Cf. A303285, A316728.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 11 2018