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 A384038 #15 May 18 2025 07:57:20
%S A384038 1,3,210,234360,4047865920,1092146608143360,4650098142288472473600,
%T A384038 314462403262051153026062745600,
%U A384038 338960040818652280796119613717033779200,5834618256563872511581456247120956565738854809600,1605370810586153268821245248112723240374305354675084328960000
%N A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.
%C A384038 Let M be a 2n X 2n matrix over GF(2) such that the column space of M is equal to the null space of M. Then M is idempotent and nullity(M) = n and index(M) = 2. If M' is similar to M then the column space of M' equals the null space of M'. Moreover, all such matrices are in the same similarity class (see Hoffman link).
%H A384038 D. G. Hoffman, <a href="https://ajc.maths.uq.edu.au/pdf/12/ocr-ajc-v12-p225.pdf">Digraphs of finite linear transformations</a>, Australasion Journal of Combinatorics,12: 225-238 (1995).
%F A384038 a(n) = A002884(n)*A006098(n).
%e A384038 a(1) = 3 because there are 3 matrices of size 2 X 2 over GF(2) with the desired property: {{0, 0}, {1, 0}}, {{0, 1}, {0, 0}}, {{1, 1}, {1, 1}}.
%t A384038 q = 2; 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]}]; Table[Product[2^(2 k) - 2^i, {i, 0, (2 k) - 1}]/aut[1, Table[2, {k}]], {k,0, 10}]
%Y A384038 Cf. A002884, A006098, A346214, A053763.
%K A384038 nonn
%O A384038 0,2
%A A384038 _Geoffrey Critzer_, May 17 2025