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%I #11 Aug 27 2014 15:42:17
%S 1,0,4,5,34,98,458,1703,7632,31222,139078,601834,2692054,12000298,
%T 54304846,246205555,1126480236,5170553126,23870651114,110597543652,
%U 514517316974,2401300629466,11243551419878,52789725198754,248514184071046,1172682575616138,5545963779654358,26282130844530698
%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), (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.: Int(Int(-2+Int(6*(1-9*x)*(1-Int(4*(1-2*x-15*x^2)^(3/2)*((1830*x^5+59*x^4+362*x^3+22*x^2-8*x+3)*(1-4*x^2)*hypergeom([7/4, 9/4],[2],64*x^3*(1+x)/(1-4*x^2)^2)+7*(750*x^4-307*x^3-213*x^2-45*x+11) *x^3*hypergeom([9/4, 11/4],[3],64*x^3*(1+x)/(1-4*x^2)^2))/((1-4*x^2)^(9/2)*(9*x-1)^2*(1+x)),x))/(1-2*x-15*x^2)^(5/2),x),x),x)/(x^2*(x-1)). - _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[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