A329066 - OEIS

%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