A277248 Number of planar walks starting at (1,1), ending at (3n,0), remaining in the first quadrant and using steps (-1,2) and (2,-1).

1, 2, 6, 24, 108, 528, 2724, 14616, 80760, 456552, 2628504, 15360216, 90879096, 543336912, 3277586136, 19924733088, 121943223576, 750756116376, 4646484480552, 28892787031008, 180420486241776, 1130930538186360, 7113550964713848, 44885329202906448

Offset: 1

Author

Feng Jishe, Oct 06 2016

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• M. Bousquet-Mélou, M. Petkovsek, Walks confined in a quadrant are not always D-finite, Theoretical Computer Science, 307(2003): 257-276.
• Ira M. Gessel, A probabilistic method for lattice path enumeration, Journal of statistical planning and inference, 14 (1986), 49-58.

Crossrefs

Cf. A048116.

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([0,3*n]):
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[{0, 3n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Dec 04 2016 after Alois P. Heinz *)

Formula

a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.06045583689606517807688682344735167414726208387456561322459238109992522838... . - Vaclav Kotesovec, Oct 07 2016

Extensions

More terms from Alois P. Heinz, Oct 06 2016