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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).

%I A277248 #30 Dec 04 2016 12:32:17
%S A277248 1,2,6,24,108,528,2724,14616,80760,456552,2628504,15360216,90879096,
%T A277248 543336912,3277586136,19924733088,121943223576,750756116376,
%U A277248 4646484480552,28892787031008,180420486241776,1130930538186360,7113550964713848,44885329202906448
%N 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).
%H A277248 Alois P. Heinz, <a href="/A277248/b277248.txt">Table of n, a(n) for n = 1..1000</a>
%H A277248 M. Bousquet-Mélou, M. Petkovsek, <a href="http://dx.doi.org/10.1016/S0304-3975(03)00219-6">Walks confined in a quadrant are not always D-finite</a>, Theoretical Computer Science, 307(2003): 257-276.
%H A277248 Ira M. Gessel, <a href="http://dx.doi.org/10.1016/0378-3758(86)90009-1">A probabilistic method for lattice path enumeration</a>, Journal of statistical planning and inference, 14 (1986), 49-58.
%F A277248 a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.06045583689606517807688682344735167414726208387456561322459238109992522838... . - _Vaclav Kotesovec_, Oct 07 2016
%p A277248 b:= proc(l) option remember; `if`(l=[1$2], 1, add((p->
%p A277248       `if`(p[1]<0, 0, b(p)))(sort((l-x))), x=[[-1, 2], [2, -1]]))
%p A277248     end:
%p A277248 a:= n-> b([0,3*n]):
%p A277248 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 06 2016
%t A277248 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_ *)
%Y A277248 Cf. A048116.
%K A277248 nonn,walk
%O A277248 1,2
%A A277248 _Feng Jishe_, Oct 06 2016
%E A277248 More terms from _Alois P. Heinz_, Oct 06 2016