%I #9 Sep 30 2021 11:42:01
%S 1,1,1,1,4,1,1,9,9,1,1,19,56,19,1,1,33,289,289,33,1,1,55,1358,4836,
%T 1358,55,1,1,85,5771,80605,80605,5771,85,1,1,128,22594,1271870,
%U 5525686,1271870,22594,128,1,1,183,81802,18478460,372302962,372302962,18478460,81802,183,1
%N Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_5)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).
%C Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.
%C 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.
%H Álvar Ibeas, <a href="/A347972/b347972.txt">Entries up to T(12, 5)</a>
%H H. Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry classes of codes</a>
%H Álvar Ibeas, <a href="/A347972/a347972.txt">Column k=1 up to n=100</a>
%H Álvar Ibeas, <a href="/A347972/a347972_1.txt">Column k=2 up to n=100</a>
%H Álvar Ibeas, <a href="/A347972/a347972_2.txt">Column k=3 up to n=100</a>
%H Álvar Ibeas, <a href="/A347972/a347972_3.txt">Column k=4 up to n=100</a>
%H Álvar Ibeas, <a href="/A347972/a347972_4.txt">Column k=5 up to n=100</a>
%F T(n, 1) = T(n - 1, 1) + A008610(n).
%e Triangle begins:
%e k: 0 1 2 3 4 5 6
%e -------------------------------
%e n=0: 1
%e n=1: 1 1
%e n=2: 1 4 1
%e n=3: 1 9 9 1
%e n=4: 1 19 56 19 1
%e n=5: 1 33 289 289 33 1
%e n=6: 1 55 1358 4836 1358 55 1
%e There are 6 = A022169(2, 1) one-dimensional subspaces in (F_5)^2. By coordinate swap, <(0, 1)> is identified with <(1, 0)> and <(1, 2)> with <(1, 3)>, while <(1, 1)> and <(1, 4)> rest invariant. Hence, T(2, 1) = 4.
%Y Cf. A022169, A008610, A241926.
%K nonn,tabl
%O 0,5
%A _Álvar Ibeas_, Sep 21 2021