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%I #20 Dec 04 2016 13:57:03
%S 1,5,70,1335,29722,726726,18924180,515622393,14537063970,420926828322,
%T 12451967363004,374889010764450,11453678709964092,354319982375393300,
%U 11078755353095301720,349637888810064314295,11124472891130365079250,356502250246211571438090
%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 2 n steps taken from {(-1, -1), (-1, 0), (-1, 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, 2008.
%F a(n) = ((1-n)*A026945(n)+(n+2)*A026945(n+1)) * (n+2)*binomial(2*n+3,n+1)/(3*(4*n+2)*(4*n+5)). [_Mark van Hoeij_, Jul 14 2010]
%p G := Int(Int(2*((12*t^2+1)*hypergeom([1/4, 3/4],[1],64*t^2/(12*t^2+1)^2)
%p -12*t^2*hypergeom([3/4,5/4],[2],64*t^2/(12*t^2+1)^2))/((1-36*t^2)*(12*t^2+1)^(3/2)),t),t)/t^2;
%p ogf := subs(t=x^(1/2), series(G, t=0, 40)); # _Mark van Hoeij_, Aug 20 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[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]
%K nonn,walk
%O 0,2
%A _Manuel Kauers_, Nov 18 2008
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