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%I #38 May 06 2021 03:17:07
%S 1,1,0,1,2,0,1,4,6,0,1,6,44,20,0,1,8,146,580,70,0,1,10,344,4332,8092,
%T 252,0,1,12,670,18152,135954,116304,924,0,1,14,1156,55252,1012664,
%U 4395456,1703636,3432,0,1,16,1834,137292,4816030,58199208,144840476,25288120,12870,0
%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=1..k} x_j^(2*j-1) + x_j^(-(2*j-1)) )^(2*n).
%H Seiichi Manyama, <a href="/A329020/b329020.txt">Antidiagonals n = 0..50, flattened</a>
%F T(n,k) = Sum_{j=0..floor((2*k-1)*n/(2*k))} (-1)^j * binomial(2*n,j) * binomial((2*k+1)*n-2*k*j-1,(2*k-1)*n-2*k*j) for k > 0.
%e (x^3 + x + 1/x + 1/x^3)^2 = x^6 + 2*x^4 + 3*x^2 + 4 + 3/x^2 + 2/x^4 + 1/x^6. So T(1,2) = 4.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 6, 8, 10, ...
%e 0, 6, 44, 146, 344, 670, ...
%e 0, 20, 580, 4332, 18152, 55252, ...
%e 0, 70, 8092, 135954, 1012664, 4816030, ...
%e 0, 252, 116304, 4395456, 58199208, 432457640, ...
%t T[n_, 0] = Boole[n == 0]; T[n_, k_] := Sum[(-1)^j * Binomial[2*n, j] * Binomial[(2*k + 1)*n - 2*k*j - 1, (2*k - 1)*n - 2*k*j], {j, 0, Floor[(2*k - 1)*n/(2*k)]}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 06 2021 *)
%Y Columns k=0-3 give A000007, A000984, A005721, A063419.
%Y Rows n=0-2 give A000012, A005843, 2*A143166.
%Y Main diagonal gives A329021.
%Y Cf. A077042.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Nov 02 2019