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A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.

%I A328714 #21 Oct 26 2019 16:05:05
%S A328714 1,1,9,25,217,921,7761,41889,345465,2162617,17605249,121120209,
%T A328714 980612161,7174425025,58079091513,442755733065,3595708057785,
%U A328714 28197440412345,230133477721665,1841288167473105,15113407062476817,122714906949538257,1013127345082389513
%N A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.
%C A328714 a(n) is the number of n-step closed walks (from origin to origin) in 4-dimensional lattice where each step changes at most one component by -1 or by +1. - _Alois P. Heinz_, Oct 26 2019
%H A328714 Alois P. Heinz, <a href="/A328714/b328714.txt">Table of n, a(n) for n = 0..1054</a>
%F A328714 From _Vaclav Kotesovec_, Oct 26 2019: (Start)
%F A328714 Recurrence: n^4*a(n) = (5*n^4 - 10*n^3 + 10*n^2 - 5*n + 1)*a(n-1) + (n-1)^2*(70*n^2 - 140*n + 113)*a(n-2) - (n-2)*(n-1)*(230*n^2 - 690*n + 583)*a(n-3) - 789*(n-3)*(n-2)^2*(n-1)*a(n-4) + 945*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
%F A328714 a(n) ~ 9^(n+2) / (16 * Pi^2 * n^2). (End)
%F A328714 E.g.f.: exp(x) * BesselI(0,2*x)^4. - _Ilya Gutkovskiy_, Oct 26 2019
%o A328714 (PARI) {a(n) = polcoef(polcoef(polcoef(polcoef((1+w+x+y+z+1/w+1/x+1/y+1/z)^n, 0), 0), 0), 0)}
%Y A328714 Row 4 of A328718.
%Y A328714 Cf. A039699.
%K A328714 nonn
%O A328714 0,3
%A A328714 _Seiichi Manyama_, Oct 26 2019
%E A328714 a(19)-a(22) from _Alois P. Heinz_, Oct 26 2019