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%I #54 Apr 02 2021 02:46:35
%S 1,4,1,36,12,1,400,588,20,1,4900,49440,2100,28,1,63504,5187980,423440,
%T 4956,36,1,853776,597027312,117234740,1751680,9540,44,1,11778624,
%U 71962945824,36938855520,907687900,5101200,16236,52,1
%N 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=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*k).
%C T(n,k) is the number of (2*k)-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 T(n,k) is the 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*k).
%H Seiichi Manyama, <a href="/A329066/b329066.txt">Antidiagonals n = 0..50, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%F See the second code written in PARI.
%e Square array begins:
%e 1, 4, 36, 400, 4900, ...
%e 1, 12, 588, 49440, 5187980, ...
%e 1, 20, 2100, 423440, 117234740, ...
%e 1, 28, 4956, 1751680, 907687900, ...
%e 1, 36, 9540, 5101200, 4190017860, ...
%o (PARI) {T(n, k) = polcoef(polcoef((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*k), 0), 0)}
%o (PARI) f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
%o T(n, k) = sum(j=0, 2*k, (-1)^j*binomial(2*k, j)*polcoef(f(n)^j*f(n-1)^(2*k-j), 0)^2)
%Y Columns k=0-1 give A000012, A017113.
%Y Rows n=0-2 give A002894, A329024, A329067.
%Y Main diagonal gives A342964.
%Y Cf. A005408, A328718, A329074, A329078.
%K nonn,tabl,walk
%O 0,2
%A _Seiichi Manyama_, Nov 03 2019
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