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A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.

%I A329074 #40 Nov 05 2019 14:09:42
%S A329074 1,1,1,1,0,1,1,8,0,1,1,24,16,0,1,1,216,48,24,0,1,1,1200,1200,72,32,0,
%T A329074 1,1,8840,10200,3336,96,40,0,1,1,58800,165760,34800,7008,120,48,0,1,1,
%U A329074 423640,2032800,912840,82800,12600,144,56,0,1
%N A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.
%C A329074 T(n,k) is the number of k-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n).
%C A329074 T(n,k) is the constant term in the expansion of (Sum_{j=0..2*n} (x^j + 1/x^j)*(y^(2*n-j) + 1/y^(2*n-j)) - x^(2*n) - 1/x^(2*n) - y^(2*n) - 1/y^(2*n))^k for n > 0.
%H A329074 Seiichi Manyama, <a href="/A329074/b329074.txt">Antidiagonals n = 0..100, flattened</a>
%H A329074 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%F A329074 T(0,k) = 1^k = 1.
%F A329074 See the second code written in PARI.
%e A329074 Square array begins:
%e A329074    1, 1,  1,   1,     1,      1, ...
%e A329074    1, 0,  8,  24,   216,   1200, ...
%e A329074    1, 0, 16,  48,  1200,  10200, ...
%e A329074    1, 0, 24,  72,  3336,  34800, ...
%e A329074    1, 0, 32,  96,  7008,  82800, ...
%e A329074    1, 0, 40, 120, 12600, 162000, ...
%o A329074 (PARI) {T(n, k) = if(n==0, 1, polcoef(polcoef((sum(j=0, 2*n, (x^j+1/x^j)*(y^(2*n-j)+1/y^(2*n-j)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^k, 0), 0))}
%o A329074 (PARI) f(n) = (x^(n+1)-1/x^n)/(x-1);
%o A329074 T(n, k) = if(n==0, 1, sum(j=0, k, (-1)^(k-j)*binomial(k, j)*polcoef(f(n)^j*f(n-1)^(k-j), 0)^2))
%Y A329074 Rows n=0-3 give A000012, A094061, A329075, A329077.
%Y A329074 Main diagonal gives A329076.
%Y A329074 Cf. A329066.
%K A329074 nonn,tabl,walk
%O A329074 0,8
%A A329074 _Seiichi Manyama_, Nov 03 2019