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A316727
Number of permutations of {0,1,...,2n} with first element n whose sequence of ascents and descents forms a Dyck path.
3
1, 1, 5, 87, 3337, 223333, 23068057, 3403720071, 679894572497, 176710079709345, 57967294285022281, 23427042148948682599, 11437832700333350250001, 6637473822604173137681381, 4515971399162518697397538173, 3560540787622773257563653593551
OFFSET
0,3
COMMENTS
All terms are odd.
LINKS
FORMULA
a(n) = A316728(n,n).
a(n) ~ c * 4^n * (n!)^2 / n^2, where c = 0.47441051698109564449415497840875665319801746745142596395217012466627... - Vaclav Kotesovec, Jul 15 2018
EXAMPLE
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.
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 *)
CROSSREFS
Sequence in context: A220617 A231702 A069948 * A216088 A305001 A324092
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 11 2018
STATUS
approved