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%I #10 Jan 27 2022 21:40:39
%S 1,3,11,43,175,731,3111,13427,58591,257947,1143943,5104419,22896303,
%T 103169899,466725143,2118787187,9648585791,44060516667,201709358631,
%U 925531659971,4255568177615,19604179972363,90468636882231,418164385032723,1935725673812575,8973094439246811,41648668456569671
%N Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, -1), (-1, -1, 0), (0, 0, 1), (0, 1, 0), (1, 1, 1)}.
%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008-2009.
%H Xiang-Ke Chang, X.-B. Hu, H. Lei, Y.-N. Yeh, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p8">Combinatorial proofs of addition formulas</a>, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
%t aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, -1 + j, -1 + k, -1 + n] + aux[i, -1 + j, k, -1 + n] + aux[i, j, -1 + k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n] + aux[1 + i, 1 + j, 1 + k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]
%Y Cf. A151292 (similar in N^2).
%K nonn,walk
%O 0,2
%A _Manuel Kauers_, Nov 18 2008
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