OFFSET
0,5
COMMENTS
Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.
Regarding the formula for column k = 1, note that A241926(q - 1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.
LINKS
Álvar Ibeas, Entries up to T(14, 6)
H. Fripertinger, Isometry classes of codes
H. Fripertinger, Number of the isometry classes of all quaternary (n,k)-codes
Álvar Ibeas, Column k=1 up to n=100
Álvar Ibeas, Column k=2 up to n=100
Álvar Ibeas, Column k=3 up to n=100
Álvar Ibeas, Column k=4 up to n=100
Álvar Ibeas, Column k=5 up to n=100
Álvar Ibeas, Column k=6 up to n=100
FORMULA
T(n, 1) = T(n - 1, 1) + A007997(n + 5).
EXAMPLE
Triangle begins:
k: 0 1 2 3 4 5 6
-------------------------------
n=0: 1
n=1: 1 1
n=2: 1 3 1
n=3: 1 7 7 1
n=4: 1 12 31 12 1
n=5: 1 19 111 111 19 1
n=6: 1 29 361 964 361 29 1
There are 5 = A022168(2, 1) one-dimensional subspaces in (F_4)^2, namely, those generated by vectors (0, 1), (1, 0), (1, 1), (1, x), and (1, x + 1), where F_4 = F_2[x] / (x^2 + x + 1). The coordinate swap identifies the first two on the one hand and the last two on the other, while <(1, 1)> is invariant. Hence, T(2, 1) = 3.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Álvar Ibeas, Sep 21 2021
STATUS
approved