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A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^n.

%I A329077 #20 Nov 05 2019 14:05:26
%S A329077 1,0,24,72,3336,34800,912840,15661520,355423880,7241240160,
%T A329077 160151370624,3461028611040,76789098028104,1700195813892576,
%U A329077 38037857914721808,853169553940415712,19240825799184080520,435267116844063531456,9882232970998312871232
%N A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^n.
%C A329077 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| = 6).
%H A329077 Seiichi Manyama, <a href="/A329077/b329077.txt">Table of n, a(n) for n = 0..500</a> (terms 0..230 from Vaclav Kotesovec)
%H A329077 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%F A329077 Conjecture: a(n) ~ 6 * 24^n / (73*Pi*n). - _Vaclav Kotesovec_, Nov 04 2019
%o A329077 (PARI) {a(n) = polcoef(polcoef((sum(k=-3, 3, x^k)*sum(k=-3, 3, y^k)-sum(k=-2, 2, x^k)*sum(k=-2, 2, y^k))^n, 0), 0)}
%o A329077 (PARI) {a(n) = polcoef(polcoef((sum(k=0, 6, (x^k+1/x^k)*(y^(6-k)+1/y^(6-k)))-x^6-1/x^6-y^6-1/y^6)^n, 0), 0)}
%o A329077 (PARI) f(n) = (x^(n+1)-1/x^n)/(x-1);
%o A329077 a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(3)^k*f(2)^(n-k), 0)^2)
%Y A329077 Row n=3 of A329074.
%Y A329077 Cf. A002894, A094061, A329024, A329067, A329075.
%K A329077 nonn,walk
%O A329077 0,3
%A A329077 _Seiichi Manyama_, Nov 04 2019