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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of (1 + x_1 + x_2 + ... + x_n + 1/x_1 + 1/x_2 + ... + 1/x_n)^k.
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%I #35 Oct 30 2019 08:28:33

%S 1,1,1,1,1,1,1,3,1,1,1,7,5,1,1,1,19,13,7,1,1,1,51,61,19,9,1,1,1,141,

%T 221,127,25,11,1,1,1,393,1001,511,217,31,13,1,1,1,1107,4145,3301,921,

%U 331,37,15,1,1,1,3139,18733,16297,7761,1451,469,43,17,1,1

%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 (1 + x_1 + x_2 + ... + x_n + 1/x_1 + 1/x_2 + ... + 1/x_n)^k.

%C T(n,k) is the number of k-step closed walks (from origin to origin) in n-dimensional lattice where each step changes at most one component by -1 or by +1. - _Alois P. Heinz_, Oct 26 2019

%C Conjecture: Row r is asymptotic to (2*r+1)^(n + r/2) / (2^r * (Pi*n)^(r/2)). - _Vaclav Kotesovec_, Oct 27 2019

%H Alois P. Heinz, <a href="/A328718/b328718.txt">Antidiagonals n = 0..140, flattened</a>

%F From _Vaclav Kotesovec_, Oct 30 2019: (Start)

%F Columns:

%F T(n,2) = 2*n + 1.

%F T(n,3) = 6*n + 1.

%F T(n,4) = 12*n^2 + 6*n + 1.

%F T(n,5) = 60*n^2 - 10*n + 1.

%F T(n,6) = 120*n^3 + 20*n + 1.

%F T(n,7) = 840*n^3 - 840*n^2 + 392*n + 1. (End)

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 3, 7, 19, 51, 141, 393, ...

%e 1, 1, 5, 13, 61, 221, 1001, 4145, ...

%e 1, 1, 7, 19, 127, 511, 3301, 16297, ...

%e 1, 1, 9, 25, 217, 921, 7761, 41889, ...

%e 1, 1, 11, 31, 331, 1451, 15101, 85961, ...

%e 1, 1, 13, 37, 469, 2101, 26041, 153553, ...

%Y Rows n=0-5 give A000012, A002426, A201805, A328713, A328714, A328715.

%Y Main diagonal is A328716.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Oct 26 2019