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A149424 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, 0, 0), (0, -1, 0), (0, 0, -1), (1, 1, 1)}.

%I A149424 #10 Jul 12 2021 15:27:05
%S A149424 1,1,4,13,40,136,496,1753,6256,22912,85216,314836,1170688,4396048,
%T A149424 16623328,62744017,237680992,904962400,3459831424,13219219972,
%U A149424 50621972224,194465172304,749061374848,2884682636764,11126422372864,43007603099296,166555051934848,644984620465264,2500560314630656
%N A149424 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, 0, 0), (0, -1, 0), (0, 0, -1), (1, 1, 1)}.
%H A149424 Alois P. Heinz, <a href="/A149424/b149424.txt">Table of n, a(n) for n = 0..300</a>
%H A149424 A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, <a href="http://arxiv.org/abs/0811.2899">ArXiv 0811.2899</a>.
%F A149424 a(n) == 1 (mod 3). - _Alois P. Heinz_, Jul 12 2021
%p A149424 b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+
%p A149424       add(`if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..3))
%p A149424     end:
%p A149424 a:= n-> b(n, [0$3]):
%p A149424 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 26 2021
%t A149424 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, j, 1 + k, -1 + n] + aux[i, 1 + j, k, -1 + n] + aux[1 + i, j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]
%Y A149424 Cf. A151265.
%Y A149424 Column k=3 of A335570.
%K A149424 nonn,walk
%O A149424 0,3
%A A149424 _Manuel Kauers_, Nov 18 2008