A328716 -id:A328716 - cp's OEIS frontend

A328718 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.

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, 221, 127, 25, 11, 1, 1, 1, 393, 1001, 511, 217, 31, 13, 1, 1, 1, 1107, 4145, 3301, 921, 331, 37, 15, 1, 1, 1, 3139, 18733, 16297, 7761, 1451, 469, 43, 17, 1, 1

Offset: 0

Seiichi Manyama, Oct 26 2019

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

Conjecture: Row r is asymptotic to (2*r+1)^(n + r/2) / (2^r * (Pi*n)^(r/2)). - Vaclav Kotesovec, Oct 27 2019

			Square array begins:
   1, 1,  1,  1,   1,    1,     1,      1, ...
   1, 1,  3,  7,  19,   51,   141,    393, ...
   1, 1,  5, 13,  61,  221,  1001,   4145, ...
   1, 1,  7, 19, 127,  511,  3301,  16297, ...
   1, 1,  9, 25, 217,  921,  7761,  41889, ...
   1, 1, 11, 31, 331, 1451, 15101,  85961, ...
   1, 1, 13, 37, 469, 2101, 26041, 153553, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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

Main diagonal is A328716.

From Vaclav Kotesovec, Oct 30 2019: (Start)
Columns:
T(n,2) = 2*n + 1.
T(n,3) = 6*n + 1.
T(n,4) = 12*n^2 + 6*n + 1.
T(n,5) = 60*n^2 - 10*n + 1.
T(n,6) = 120*n^3 + 20*n + 1.
T(n,7) = 840*n^3 - 840*n^2 + 392*n + 1. (End)

A342964 Constant term in the expansion of ( (Sum_{j=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*n).

Original entry on oeis.org

1, 12, 2100, 1751680, 4190017860, 20874801722544, 177661172742061008, 2295966445175463883680, 41848194615009705993547620, 1022849138778659709119846990032, 32304962696573489860535097887683296

Offset: 0

Seiichi Manyama, Mar 31 2021

nonn

Number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n+1).

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*n).

Wikipedia, Taxicab geometry.

Main diagonal of A329066.

Cf. A002894, A328716, A329024, A329067, A329076.

f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
a(n) = sum(j=0, 2*n, (-1)^j*binomial(2*n, j)*polcoef(f(n)^j*f(n-1)^(2*n-j), 0)^2);

cp's OEIS Frontend

A328718 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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A342964 Constant term in the expansion of ( (Sum_{j=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*n).

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cp's OEIS Frontend

A328718 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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Author

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Examples

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Crossrefs

Formula

A342964 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*n).

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PARI

A342964 Constant term in the expansion of ( (Sum_{j=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*n).