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A260774 Certain directed lattice paths.

%I A260774 #22 Jul 15 2022 04:32:36
%S A260774 1,6,33,189,1107,6588,39663,240894,1473147,9058554,55954395,346934745,
%T A260774 2157989445,13459891500,84152389833,527224251861,3309194474451,
%U A260774 20804569738218,130987600581699,825796890644895,5212349717906889,32935490120006604,208316726580941037
%N A260774 Certain directed lattice paths.
%C A260774 See Dziemianczuk (2014) for precise definition.
%H A260774 Alois P. Heinz, <a href="/A260774/b260774.txt">Table of n, a(n) for n = 0..1235</a> (first 101 terms from Lars Blomberg)
%H A260774 M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%F A260774 See Dziemianczuk (2014) Equation (33a) with m=1.
%F A260774 From _Vaclav Kotesovec_, Jul 15 2022: (Start)
%F A260774 Recurrence: (n+1)*(4*n - 3)*a(n) = 6*(4*n^2 - n - 1)*a(n-1) + 3*(n-1)*(4*n + 1)*a(n-2).
%F A260774 a(n) ~ (3 + 2*sqrt(3))^(n+1) / sqrt(6*Pi*n). (End)
%p A260774 b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
%p A260774       `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
%p A260774       `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
%p A260774     end:
%p A260774 a:= n-> b(n, 1):
%p A260774 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 21 2021
%t A260774 b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
%t A260774      If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
%t A260774      If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
%t A260774 a[n_] := b[n, 1];
%t A260774 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 02 2022, after _Alois P. Heinz_ *)
%Y A260774 Cf. A122868, A064641, A156016.
%Y A260774 Cf. A006139, A179191, A052709, A025227.
%K A260774 nonn
%O A260774 0,2
%A A260774 _N. J. A. Sloane_, Jul 30 2015
%E A260774 More terms from _Lars Blomberg_, Aug 01 2015