This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379105 #44 Jan 03 2025 09:37:22
%S A379105 1,1,1,6,9,1,168,294,49,1,20160,37800,7350,225,1,9999360,19373760,
%T A379105 4036200,144150,961,1,20158709760,39687459840,8543828160,326932200,
%U A379105 2542806,3969,1,163849992929280,325139829719040,71124337751040,2812314375360,23435953128,42677334,16129,1
%N A379105 Triangular array read by rows. T(n,k) is the number of n X n matrices T over GF(2) such that there are exactly 2^k vectors v in GF(2)^n with Tv=v, n>=0, 0<=k<=n.
%C A379105 Sum_{k=0..n} T(n,k)*2^k = (2^(n+1)-1)*2^(n^2-n) so that as n->oo the average number of fixed points is 2.
%C A379105 T(n,k) is also the number of n X n matrices over GF(2) with nullity k. As n->oo, the probability that a random n X n matrix over GF(q) has nullity k is 1/|GL_k(F_q)|*Product_{i>=k+1} (1 - 1/q^i). - _Geoffrey Critzer_, Dec 31 2024
%F A379105 T(n,k)=Product_{j=0..n-k-1} (2^n - 2^j)^2/(2^(n-k)-2^j). - _Geoffrey Critzer_, Dec 31 2024
%e A379105 Triangle T(n,k) begins:
%e A379105 1;
%e A379105 1, 1;
%e A379105 6, 9, 1;
%e A379105 168, 294, 49, 1;
%e A379105 20160, 37800, 7350, 225, 1;
%e A379105 9999360, 19373760, 4036200, 144150, 961, 1;
%e A379105 ...
%t A379105 nn = 5; b[p_, i_] := Count[p, i];d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] :=Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; L=Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]; g[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &, L]]; Map[Select[#, # > 0 &] &, Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0,nn}] CoefficientList[Series[g[u, 1, 1] g[u, v, 1] Product[g[u, 1, deg]^\[Nu][[deg]], {deg, 2, nn}], {u, 0, nn}], {u,v}]] // Grid
%Y A379105 Cf. A060867 (T(n,n-1)), A002884 (column k=0), A086699 (column k=1), A346381, A286331.
%Y A379105 Row sums give A002416.
%K A379105 nonn,tabl
%O A379105 0,4
%A A379105 _Geoffrey Critzer_, Dec 15 2024