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A342964 Constant term in the expansion of ( (Sum_{j=0..n} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n} y^(2*j+1)+1/y^(2*j+1)) - (Sum_{j=0..n-1} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n-1} y^(2*j+1)+1/y^(2*j+1)) )^(2*n).

%I A342964 #16 Apr 01 2021 14:41:22
%S A342964 1,12,2100,1751680,4190017860,20874801722544,177661172742061008,
%T A342964 2295966445175463883680,41848194615009705993547620,
%U A342964 1022849138778659709119846990032,32304962696573489860535097887683296
%N A342964 Constant term in the expansion of ( (Sum_{j=0..n} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n} y^(2*j+1)+1/y^(2*j+1)) - (Sum_{j=0..n-1} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n-1} y^(2*j+1)+1/y^(2*j+1)) )^(2*n).
%C A342964 Number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n+1).
%C A342964 Constant term in the expansion of (Sum_{j=0..2*n+1} (x^j + 1/x^j)*(y^(2*n+1-j) + 1/y^(2*n+1-j)) - x^(2*n+1) - 1/x^(2*n+1) - y^(2*n+1) - 1/y^(2*n+1))^(2*n).
%H A342964 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%o A342964 (PARI) f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
%o A342964 a(n) = sum(j=0, 2*n, (-1)^j*binomial(2*n, j)*polcoef(f(n)^j*f(n-1)^(2*n-j), 0)^2);
%Y A342964 Main diagonal of A329066.
%Y A342964 Cf. A002894, A328716, A329024, A329067, A329076.
%K A342964 nonn
%O A342964 0,2
%A A342964 _Seiichi Manyama_, Mar 31 2021