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 A151472 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), (0, -1), (0, 1), (1, -1), (1, 0)}. 0

%I #11 Dec 04 2016 13:57:03

%S 1,1,3,9,30,110,423,1687,6984,29574,128074,564652,2527292,11463972,

%T 52602015,243824807,1140448152,5377337150,25539196048,122093592944,

%U 587170555168,2839207157456,13797304069674,67357039620092,330225541717108,1625329978935340,8028874036140468,39796190100237612

%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), (0, -1), (0, 1), (1, -1), (1, 0)}.

%H M. Bousquet-MÃ©lou 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(Int(x*(-36-2*Int((1-4*x-12*x^2)^(3/2)*((256*x^5+416*x^4+128*x^3+3*x+3)*hypergeom([5/4, 7/4],[1],64*x^3*(2*x+1)/(8*x^2-1)^2)-7*x*(40*x^4+68*x^3-4*x^2-18*x-3)*hypergeom([5/4, 11/4],[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^3. - _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, 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, 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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