A300954 Number of Dyck paths whose sequence of ascent lengths is exactly n+1, n+2, ..., 2n.

1, 1, 3, 26, 425, 10647, 365512, 16067454, 864721566, 55202528425, 4083666929771, 343854336973368, 32493430569907125, 3406873823160467912, 392619681705581846700, 49342834390595374213214, 6717520607597479710109299, 984991858956314599670220717, 154785386247352261724279606367

Offset: 0

Author

Alois P. Heinz, Mar 16 2018

Keywords

• nonn

Comments

Dyck paths counted by a(n) have semilength (3*n^2 + n)/2 = A005449(n) and length A049451(n).

Examples

			a(0) = 1: the empty path.
a(1) = 1: uudd.
a(2) = 3: uuuduuuudddddd, uuudduuuuddddd, uuuddduuuudddd.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150
• Wikipedia, Counting lattice paths

Crossrefs

Main diagonal of A107876.

Cf. A005449, A049451.

Programs

Maple

a:= proc(m) option remember; local b; b:=
      proc(n, i) option remember; `if`(i>=2*m, 1,
        add(b(n+i-j, i+1), j=1..n+i))
      end; b(0, m+1)
    end:
seq(a(n), n=0..20);

Mathematica

a[m_] := a[m] = Module[{b}, b[n_, i_] := b[n, i] = If[i >= 2m, 1, Sum[b[n + i - j, i + 1], {j, 1, n + i}]]; b[0, m + 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A107876(2n,n).