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%I #24 Dec 04 2016 13:57:04
%S 1,1,4,12,54,210,1020,4445,22610,105210,551376,2678676,14332164,
%T 71788860,389991888,1998530820,10984120290,57293297490,317798892840,
%U 1681213457352,9395215622364,50278804820244,282711201927336,1527524202392370,8633634028624332,47028406025950300,266981514041485600
%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, 1), (0, -1), (0, 1), (1, -1), (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.
%F G.f.: Int(Int(((3-6*x)*hypergeom([1/2,3/2],[1],16*x/(12*x^2+8*x+1))+(6*x-1)*hypergeom([1/2,3/2],[2],16*x/(12*x^2+8*x+1)))/(12*x^2+8*x+1)^(3/2),x),x)/x^2. - _Mark van Hoeij_, Aug 25 2014
%F a(n) = A001006(n) * binomial(n,floor(n/2)). - _Benedict W. J. Irwin_, Oct 14 2016
%p seq(binomial(n,floor(n/2))*add(n!/((n-2*k)!*k!*(k+1)!), k=0..floor(n/2)), n=0..26); # _Mark van Hoeij_, May 12 2013
%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, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]
%Y Cf. A001006 (Motzkin numbers).
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008
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