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A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.

%I A329075 #31 Nov 05 2019 14:06:56
%S A329075 1,0,16,48,1200,10200,165760,2032800,30115120,417189360,6116225976,
%T A329075 88579001280,1308168101856,19335388664592,288264711738432,
%U A329075 4311842765438208,64819095869951280,977630677389002208,14796595755047824432,224583060859608559680,3417918348978709970680
%N A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.
%C A329075 Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 4).
%H A329075 Seiichi Manyama, <a href="/A329075/b329075.txt">Table of n, a(n) for n = 0..500</a> (terms 0..300 from Vaclav Kotesovec)
%H A329075 Vaclav Kotesovec, <a href="/A329075/a329075.txt">Recurrence of order 8 (conjectured)</a>
%H A329075 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%F A329075 Conjecture: a(n) ~ 2 * 16^n / (11*Pi*n). - _Vaclav Kotesovec_, Nov 04 2019
%o A329075 (PARI) {a(n) = polcoef(polcoef((sum(k=-2, 2, x^k)*sum(k=-2, 2, y^k)-(x+1+1/x)*(y+1+1/y))^n, 0), 0)}
%o A329075 (PARI) {a(n) = polcoef(polcoef((sum(k=0, 4, (x^k+1/x^k)*(y^(4-k)+1/y^(4-k)))-x^4-1/x^4-y^4-1/y^4)^n, 0), 0)}
%o A329075 (PARI) f(n) = (x^(n+1)-1/x^n)/(x-1);
%o A329075 a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(2)^k*f(1)^(n-k), 0)^2)
%Y A329075 Row n=2 of A329074.
%Y A329075 Cf. A094061, A329067.
%K A329075 nonn,walk
%O A329075 0,3
%A A329075 _Seiichi Manyama_, Nov 03 2019