OFFSET
0,8
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
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
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)
EXAMPLE
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, ...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 26 2019
STATUS
approved