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A328426 Number of 2n-step walks on cubic lattice starting at (0,0,0), ending at (0,n,n) and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

%I A328426 #13 May 13 2020 07:02:43
%S A328426 1,4,72,1480,33880,819504,20562696,529399728,13892650200,369988864960,
%T A328426 9969694635472,271217782469088,7436910593958664,205289086309888000,
%U A328426 5699233814774374800,159004177074207384480,4455230052707022818520,125307788218001246547360
%N A328426 Number of 2n-step walks on cubic lattice starting at (0,0,0), ending at (0,n,n) and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).
%C A328426 These walks are not restricted to the first (nonnegative) octant.
%H A328426 Alois P. Heinz, <a href="/A328426/b328426.txt">Table of n, a(n) for n = 0..400</a>
%F A328426 a(n) = A328347(2n,n).
%p A328426 b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(add(add(
%p A328426       `if`(i+j+k=1, (h-> `if`((w->w<0 or w<max(map(abs, h)))(add(t, t=h)),
%p A328426        0, b(h)))(sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
%p A328426     end:
%p A328426 a:= n-> b(sort([0, n$2])):
%p A328426 seq(a(n), n=0..23);
%t A328426 b[l_] := b[l] = If[Last[l] == 0, 1, Function[r, Sum[If[i + j + k == 1, Function[h, If[Function[w, w < 0 || w < Max[Map[Abs, h]]][Total[h]], 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][{-1, 0, 1}]];
%t A328426 a[n_] := b[Sort[{0, n, n}]];
%t A328426 a /@ Range[0, 23] (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y A328426 Cf. A328347.
%K A328426 nonn,walk
%O A328426 0,2
%A A328426 _Alois P. Heinz_, Oct 15 2019