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%I #28 May 11 2020 09:44:37
%S 1,2,10,65,490,4032,35244,321750,3035890,29395652,290621188,
%T 2922898706,29821640380,307994453600,3214454901480,33855533036865,
%U 359438259174930,3843173300937300,41351489731559700,447450028715934090,4866409456815200580,53171146669028038560,583400942149413843000
%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, 0), (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], 2008-2009.
%F G.f.: Int(hypergeom([1/2,3/2],[2],16*x/(1+4*x))/(1+4*x)^(1/2),x)/x. - _Mark van Hoeij_, Aug 20 2014
%F a(n) = (-1)^n*(1+n/3)*binomial(2*n+1,n)*hypergeom([5/2,-n],[4],4)/(2*n+1) = A000108(n)*(A001006(n)+A001006(n+1))*(n+3)/6. - _Mark van Hoeij_, Aug 24 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, 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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