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A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.

%I A329076 #34 Nov 05 2019 16:26:20
%S A329076 1,0,16,72,7008,162000,17555520,1093527120,140846184640,
%T A329076 16016249944800,2550757928818680,419682645514181280,
%U A329076 82389928294166805312,17418502084657134228768,4123280170924828458697152,1054943518137131171386437600,293933660095874311773617934720,87968971083026619734709639853632
%N A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.
%C A329076 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| = 2*n).
%H A329076 Seiichi Manyama, <a href="/A329076/b329076.txt">Table of n, a(n) for n = 0..150</a> (terms 0..53 from Vaclav Kotesovec)
%H A329076 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%F A329076 Conjecture: a(n) ~ 3 * 2^(3*n - 2) * n^(n-3) / Pi. - _Vaclav Kotesovec_, Nov 05 2019
%o A329076 (PARI) {a(n) = polcoef(polcoef((sum(k=-n, n, x^k)*sum(k=-n, n, y^k)-sum(k=-n+1, n-1, x^k)*sum(k=-n+1, n-1, y^k))^n, 0), 0)}
%o A329076 (PARI) {a(n) = polcoef(polcoef((sum(k=0, 2*n, (x^k+1/x^k)*(y^(2*n-k)+1/y^(2*n-k)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^n, 0), 0)}
%o A329076 (PARI) f(n) = (x^(n+1)-1/x^n)/(x-1);
%o A329076 a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(n)^k*f(n-1)^(n-k), 0)^2)
%Y A329076 Main diagonal of A329074.
%Y A329076 Cf. A094061, A286928.
%K A329076 nonn,walk
%O A329076 0,3
%A A329076 _Seiichi Manyama_, Nov 04 2019