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%I #8 Dec 04 2016 13:57:02
%S 1,3,23,229,2564,30874,390124,5100052,68384716,935121688,12988742454,
%T 182726814050,2597994569974,37269939754622,538769767630088,
%U 7840164781172800,114752711724862584,1688183258963366500,24948722231136735010,370204576610120406342,5513458878514541095188
%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, 1), (1, 0), (1, 1)}.
%H Alois P. Heinz, <a href="/A151393/b151393.txt">Table of n, a(n) for n = 0..200</a>
%H M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.
%p b:= proc(n, l) option remember; `if` (l[1]<0 or l[2]<0 or n<l[1], 0, `if`
%p (n=0, 1, add (b(n-1, l+d), d=[[-1, -1], [-1, 1], [1, 0], [1, 1]])))
%p end:
%p a:= n-> b(2*n, [0$2]):
%p seq (a(n), n=0..30); # _Alois P. Heinz_, Jul 19 2012
%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]]; 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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