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%I #11 Aug 27 2014 15:49:09
%S 1,0,5,5,72,158,1503,4958,38390,161170,1116376,5433892,35471944,
%T 189340318,1199984953,6790736977,42491355502,249738458994,
%U 1557129072666,9386156958774,58598281352232,359465513555718,2252350134784800,13992962318626614,88084177345901544,552491177540617338
%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, -1), (-1, 0), (-1, 1), (0, -1), (1, -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-x)*(1-Int(4*(1-2*x-35*x^2)^(3/2)*((12*x^2+1)*(-10*x^6+289*x^5-515*x^4-1802*x^3-487*x^2+8*x+6)*hypergeom([9/4, 11/4],[3],64*(x^2+x+1)*x^2/(12*x^2+1)^2)+6*(590*x^6-571*x^5-103*x^4+3079*x^3 +769*x^2-14*x-6)*x^2*hypergeom([11/4, 13/4],[4],64*(x^2+x+1)*x^2/(12*x^2+1)^2))/((12*x^2+1)^(11/2)*(x-1)^2),x))/(1-2*x-35*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[-1 + 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[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008
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