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

%I #11 Aug 27 2014 15:11:14

%S 1,1,2,7,19,61,224,771,2855,11005,41963,165661,664443,2674101,

%T 10955892,45281419,188288455,790957165,3343062477,14209485769,

%U 60792612875,261330644741,1128597807923,4896682653677,21327006074731,93233880458581,409028951228459,1800105977084221,7946053746358811

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

%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(Int((1-9*x)*(6+12*Int((1-2*x-15*x^2)^(3/2)*((5-18*x-816*x^6 -832*x^5+288*x^4+128*x^3+71*x^2)*hypergeom([5/4, 7/4],[1],64*x^3*(1+x)/ (1-4*x^2)^2)+(51*x^2+8*x-1+14*x^3-336*x^5+110*x^4-744*x^6) *hypergeom([5/4, 7/4],[2],64*x^3*(1+x)/(1-4*x^2)^2))/((1-4*x^2)^(7/2)*(1+x)^2*(1-9*x)^2),x))/(1-2*x-15*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[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