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%I #33 Apr 11 2022 10:14:20
%S 1,3,22,209,2256,26296,322696,4109131,53802868,719967204,9804170810,
%T 135438150718,1893565055948,26744778067560,381061505993160,
%U 5470780479977505,79066952734823832,1149467155656304276,16798622641884084940,246654934301978877376
%N Number of square lattice walks of length 2n starting and ending at (0,0) and avoiding the West quadrant {(i,j): i < -|j|}.
%H Alois P. Heinz, <a href="/A260154/b260154.txt">Table of n, a(n) for n = 0..830</a>
%H M. Bousquet-Mélou, <a href="http://arxiv.org/abs/1511.02111">Plane lattice walks avoiding a quadrant</a>, arXiv:1511.02111 [math.CO], 2015.
%F a(2n) = 16^n/9 * ( 3* (1/2)_n^2/ (2)_n^2 + 8 *(1/2)_n* (7/6)_n/ (2)_n/ (4/3)_n - 2 *(1/2)_n*(5/6)_n/ (2)_n/ (5/3)_n), where (a)_n is the ascending factorial (proved).
%F D-finite with recurrence n*(n-1)*(3*n+2)*(4*n-5)*(3*n+1)*(4*n-7)*(n+1)^2*a(n) -4*n*(n-1)*(4*n-1)*(2*n-1)*(4*n-7)*(54*n^3-45*n^2-49*n-10)*a(n-1) +16*(n-1)*(4*n-5)*(2*n-1)*(2*n-3)*(4*n+1)*(108*n^3-396*n^2+361*n+5)*a(n-2) -64*(6*n-11)*(4*n-1)*(6*n-13)*(2*n-1)*(2*n-3)*(4*n+1)*(-5+2*n)^2*a(n-3)=0. _Alois P. Heinz_, Nov 10 2015
%F D-finite with recurrence n*(n-1)*(3*n+2)*(3*n+1)*(n+1)^2*a(n) -4*n*(n-1)*(180*n^4-360*n^3+287*n^2-71*n+2)*a(n-1) +16*(n-1)*(1440*n^5-10080*n^4+29024*n^3-42768*n^2+31867*n-9465)*a(n-2) -64*(2*n-5)*(2880*n^5-30240*n^4+128608*n^3-277008*n^2+301706*n-132501)*a(n-3) +2048*(2*n-5)*(2*n-7)*(360*n^4-4320*n^3+19474*n^2-39156*n+29691)*a(n-4) -16384*(6*n-23)*(6*n-25)*(2*n-5)*(2*n-7)*(2*n-9)^2*a(n-5)=0. - _R. J. Mathar_, Apr 11 2022
%e When n=1, only the walks NS, EW, SN contribute.
%p a:= proc(n) option remember; `if`(n<3, [1, 3, 22][n+1],
%p (4*n*(n-1)*(4*n-1)*(54*n^3-45*n^2-49*n-10)*(2*n-1)*
%p (4*n-7)*a(n-1) -(16*(n-1))*(4*n-5)*(2*n-1)*(2*n-3)*
%p (4*n+1)*(108*n^3-396*n^2+361*n+5)*a(n-2) +(64*(6*n-11))*
%p (4*n-1)*(6*n-13)*(2*n-1)*(2*n-3)*(4*n+1)*(-5+2*n)^2*a(n-3))
%p /((3*n+2)*(4*n-5)*(3*n+1)*(4*n-7)*n*(n-1)*(n+1)^2))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 10 2015
%t a[n_] := a[n] = If[n<3, {1, 3, 22}[[n+1]], (4(54n^3 - 45n^2 - 49n - 10)(4n - 7)(n-1)(2n - 1)(4n - 1) n a[n-1] - (16(n-1)(4n - 5)(2n - 1)(2n - 3)(4n + 1)(108n^3 - 396n^2 + 361n + 5) a[n-2]) + (6n - 13)(64(6n - 11))(2n - 3) (2n - 1)(4n - 1)(4n + 1)(2n - 5)^2 a[n-3])/((3n + 2)(4n - 5)(3n + 1)(4n - 7) n(n-1)(n+1)^2)]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 04 2016 after _Alois P. Heinz_ *)
%Y Cf. A260153.
%K nonn,easy,walk
%O 0,2
%A _Mireille Bousquet-Mélou_, Nov 09 2015
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