%I #12 May 11 2020 09:44:24
%S 1,0,2,3,20,60,345,1400,7770,36876,204876,1062600,5984352,32772168,
%T 187727826,1065491427,6206538910,36123454224,213645926208,
%U 1266954939822,7593846587496,45694598654640,277066122022872,1686973019649060,10331412410444280,63524084460496480,392411097399517800
%N Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 1)}.
%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.
%F G.f.: Int(Int(2*hypergeom([3/4, 5/4],[2],64*(x^2+x+1)*x^2/(12*x^2+1)^2)/(12*x^2+1)^(3/2),x),x)/x^2. - _Mark van Hoeij_, Aug 17 2014
%t aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008