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A260155 Number of walks of length 2n on the square lattice that start and end at (0,0) and avoid the negative quadrant.

%I A260155 #24 Jul 25 2018 08:06:52
%S A260155 1,4,32,318,3530,41944,522010,6719018,88726840,1195527822,16373466714,
%T A260155 227280520316,3190715296368,45226324937400,646392346047930,
%U A260155 9305481272839662,134815491199174476,1964195875748858812,28761433275110249932,423052415434610432816
%N A260155 Number of walks of length 2n on the square lattice that start and end at (0,0) and avoid the negative quadrant.
%H A260155 Alois P. Heinz, <a href="/A260155/b260155.txt">Table of n, a(n) for n = 0..800</a>
%H A260155 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 A260155 a(n) = 4*16^n/ 3^5 * ( 3^4 *f(1/2,n)* f(1/2,n+1)/ (f(2,n) * f(2,n+1)) + 4 *(24*n^2+60*n +29)* f(1/2,n)* f(7/6,n) /(f(2,n+1) *f(4/3, n+1)) -2 *(12*n^2+30*n+5) * f(1/2,n)*f(5/6,n) /(f(2,n+1)*f(5/3,n+1)) ) where f(m,n) is the ascending factorial m*(m+1)*...*(m+n-1) (proved).
%e A260155 When n=1 the four walks are NS, EW, SN, WE.
%t A260155 f[x_, n_] := x Pochhammer[x+1, n-1];
%t A260155 a[n_] := 4 16^n/3^5 (3^4 f[1/2, n] f[1/2, n + 1]/(f[2, n] f[2, n + 1]) + 4 (24n^2 + 60n + 29) f[1/2, n] f[7/6, n]/(f[2, n + 1] f[4/3, n + 1]) - 2 (12n^2 + 30n + 5) f[1/2, n] f[5/6, n]/(f[2, n + 1] f[5/3, n + 1]));
%t A260155 Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Jul 25 2018 *)
%Y A260155 Cf. A060898 for walks starting from (0,0) but in which the final point is not prescribed.
%K A260155 nonn,easy,walk
%O A260155 0,2
%A A260155 _Mireille Bousquet-Mélou_, Nov 09 2015