A227850 - OEIS

%I #20 Apr 23 2016 11:16:08

%S 1,4,1248,5401472,114070692352,7593330670240768

%N Number of Dyck paths of semilength n*(4*n+1) in which the run length sequence is a permutation of {1,...,4*n}.

%e a(1) = 4: UUDUUUDDDD (2134), UUUDUUDDDD (3124), UUUUDDUDDD (4213), UUUUDDDUDD (4312).

%p h:= proc(n, s) option remember;

%p        `if`(n>add(sort([s[]], `>`)[i], i=1..(nops(s)+1)/2), 0,

%p        add(g(n-i, s minus {i}), i=select(x-> x<=n, s)))

%p     end:

%p g:= proc(n, s) option remember;

%p        `if`(s={}, `if`(n=0, 1, 0), add(h(n+i, s minus {i}), i=s))

%p     end:

%p a:= n-> g(0, {$1..4*n}):

%p seq(a(n), n=0..3);

%t h[n_, s_] := h[n, s] = If[n > Sum[Sort[s, Greater][[i]], {i, 1, (Length[s] + 1)/2}], 0, Sum[g[n - i, s ~Complement~ {i}], {i, Select[s, # <= n&]}] ];

%t g[n_, s_] := g[n, s] = If[s == {}, If[n == 0, 1, 0], Sum[h[n + i, s  ~Complement~ {i}], {i, s}]];

%t a[n_] := g[0, Range[4*n]];

%t Table[a[n], {n, 0, 4}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)

%Y Cf. A007742, A060005, A073410, A168238.

%K nonn,more

%O 0,2

%A _David Scambler_ and _Alois P. Heinz_, Oct 31 2013