A048116 a(n) = T(2n,n), where T is given by A048113.

1, 2, 4, 12, 36, 120, 408, 1440, 5160, 18816, 69336, 258048, 967344, 3649536, 13839504, 52715952, 201556944, 773182608, 2974442112, 11471570352, 44341125312, 171732665520, 666302137056, 2589317125824, 10076939895984, 39268487472336, 153208051192848

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• walk

Comments

Number of planar walks starting at (1,1), ending at (n,n), remaining in the first quadrant and using steps (-1,2) and (2,-1).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• M. Bousquet-Mélou and M. Petkovsek, Walks confined in a quadrant are not always D-finite

Crossrefs

Cf. A048113, A277262.

Programs

Maple

b:= proc(l) option remember; `if`(l=[1$2], 1, add((p->
      `if`(p[1]<0, 0, b(p)))(sort(l-x)), x=[[-1, 2], [2, -1]]))
    end:
a:= n-> b([n$2]):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 06 2016

Mathematica

b[l_List] := b[l] = If[l == {1, 1}, 1, Sum[Function[p, If[p[[1]] < 0, 0, b[p]]][Sort[l - x]], {x, {{-1, 2}, {2, -1}}}]]; a[n_] := b[{n, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

Formula

a(n) ~ c * 4^n / sqrt(n), where c = 0.03748220353529780423030694970938451772844604409392271... . - Vaclav Kotesovec, Oct 07 2016

Extensions

a(1)=1 prepended by Alois P. Heinz, Oct 06 2016