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%I
%S 1,0,2,2,14,28,148,408,1948,6316,29164,103336,473860,1770364,8141872,
%T 31491484,145696432,577601620,2689508560,10864037780,50889023352,
%U 208655144644,982532205120,4078460733332,19292955235136,80919972951468,384307594445168,1626307851201068,7750396627918568
%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, 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.
%F G.f.: Int(Int((450*x^3+90*x^2-42*x-2)*(1+3*Int((1-2*x-15*x^2)^(1/2)*(x-1+((16*x^2+1)*(19872*x^6+3753*x^5-5565*x^4-510*x^3+294*x^2+20*x-8)*hypergeom([5/4, 7/4],[2],64*x^2*(1+x^2)/(16*x^2+1)^2)-10*x^2*(2880*x^6-2205*x^5-2139*x^4+423*x^3+45*x^2+12*x-8)*hypergeom([7/4, 9/4],[3],64*x^2*(1+x^2)/(16*x^2+1)^2))/(16*x^2+1)^(7/2))/ (225*x^3+45*x^2-21*x-1)^2,x))/(1-2*x-15*x^2)^(3/2),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, 1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + 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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