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

Author

Seiichi Manyama, Oct 26 2019

Keywords

• nonn
• tabl

Comments

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

Examples

			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, ...
		

Links

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

Crossrefs

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

Main diagonal is A328716.

Formula

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)