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A328713 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z)^n.

%I A328713 #26 Oct 26 2019 16:05:43
%S A328713 1,1,7,19,127,511,3301,16297,103279,570367,3595177,21167917,133789789,
%T A328713 818625133,5207248879,32649752779,209258291599,1333828204303,
%U A328713 8612806088761,55546469634733,361143420408337,2349709451702737,15370341546766939,100695951740818903,662213750028892429
%N A328713 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z)^n.
%C A328713 a(n) is the number of n-step closed walks (from origin to origin) in cubic lattice where each step changes at most one component by -1 or by +1. - _Alois P. Heinz_, Oct 26 2019
%H A328713 Alois P. Heinz, <a href="/A328713/b328713.txt">Table of n, a(n) for n = 0..1189</a>
%F A328713 From _Vaclav Kotesovec_, Oct 26 2019: (Start)
%F A328713 Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) + (n-1)*(34*n^2 - 68*n + 41)*a(n-2) - 38*(n-2)*(n-1)*(2*n - 3)*a(n-3) - 105*(n-3)*(n-2)*(n-1)*a(n-4).
%F A328713 a(n) ~ 7^(n + 3/2) / (8 * Pi^(3/2) * n^(3/2)). (End)
%F A328713 E.g.f.: exp(x) * BesselI(0,2*x)^3. - _Ilya Gutkovskiy_, Oct 26 2019
%e A328713 (1+x+y+z+1/x+1/y+1/z)^2 = x^2 + 1/x^2 + y^2 + 1/y^2 + z^2 + 1/z^2 + 2 * (xy + 1/(xy) + yz + 1/(yz) + zx + 1/(zx) + x/y + y/x + y/z + z/y + z/x + x/z + x + 1/x + y + 1/y + z + 1/z) + 7. So a(2) = 7.
%o A328713 (PARI) {a(n) = polcoef(polcoef(polcoef((1+x+y+z+1/x+1/y+1/z)^n, 0), 0), 0)}
%Y A328713 Row 3 of A328718.
%Y A328713 Cf. A002426, A002896, A201805.
%K A328713 nonn
%O A328713 0,3
%A A328713 _Seiichi Manyama_, Oct 26 2019