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

%I A303286 #13 May 31 2018 12:38:01
%S A303286 1,4,60,1974,114972,10490392,1384890104,250150900354,59317740001132,
%T A303286 17886770092245360,6687689652133397064,3037468107154650475868,
%U A303286 1647659575564603380270360,1052309674407466474533397824,781725844087366504901991503920,668408235613132734111402947167658
%N A303286 Number of permutations p of [2n+1] such that the sequence of ascents and descents of 0p0 forms a Dyck path.
%H A303286 Alois P. Heinz, <a href="/A303286/b303286.txt">Table of n, a(n) for n = 0..200</a>
%H A303286 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A303286 a(n) ~ c * 2^(2*n) * n^(2*n) / exp(2*n), where c = 45.0971960423271758887353825240016439879529954831112316... - _Vaclav Kotesovec_, May 22 2018
%e A303286 a(1) = 4: 132, 213, 231, 312.
%p A303286 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A303286       `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
%p A303286       `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
%p A303286     end:
%p A303286 a:= n-> b(0, 2*n+1, 0):
%p A303286 seq(a(n), n=0..20);
%t A303286 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]];
%t A303286 a[n_] := b[0, 2*n + 1, 0];
%t A303286 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A303286 Bisection (odd part) of A303284.
%Y A303286 Cf. A180056.
%K A303286 nonn
%O A303286 0,2
%A A303286 _Alois P. Heinz_, Apr 20 2018