%I #11 Aug 27 2014 15:43:57
%S 1,1,6,20,95,432,2100,10429,52990,273872,1435464,7610704,40747432,
%T 219972284,1196042952,6543872976,36000272857,199016494848,
%U 1104987607068,6159118520824,34451516940832,193323839813568,1087995843781768,6139413903894528,34728866786674200,196895381265884724
%N Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, 0), (-1, 1), (0, -1), (0, 1), (1, 0), (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.
%F G.f.: 1+6*Int(Int(Int(x*(14+Int((1-4*x-12*x^2)^(3/2)*((-160*x^4-176*x^3-56*x^2-8*x-1)*hypergeom([5/4, 7/4],[1],64*x^3*(2*x+1)/(8*x^2-1)^2)+4*x^2*(32*x^3-10*x^2-19*x-4)*hypergeom([5/4, 7/4],[2], 64*x^3*(2*x+1)/(8*x^2-1)^2))/((2*x+1)*(1-8*x^2)^(7/2)*x^2),x))/(1-4*x-12*x^2)^(5/2),x),x),x)/x^2. - _Mark van Hoeij_, Aug 27 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, 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]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008