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 A342245 #13 Jun 20 2025 16:31:32
%S A342245 1,3,21,339,12483,1074339,219474243,107174166147,126918737362179,
%T A342245 367662330459585027,2614066808849501254659,45985259502347910886975491,
%U A342245 2009925824909891218929491103747,218411680908756813835229484489351171,59296916710446845619466630380450779971587
%N A342245 Number of ordered pairs (S,T) of n X n idempotent matrices over GF(2) such that ST = TS = S.
%C A342245 The components in the ordered pairs are not necessarily distinct.
%C A342245 The relation S<=T iff ST=TS=S gives a partial ordering on the idempotent matrices enumerated in A132186. Each length k chain (from bottom to top) in the poset corresponds to an ordered direct sum decomposition of GF(2)^n into exactly k subspaces.
%F A342245 Let E(x) = Sum_{n>=0} x^n/(2^binomial(n,2) * [n]_2!) where [n]_2! = A005329(n). Then E(x)^3 = Sum_{n>=0} a(n)x^n/(2^binomial(n,2) * [n]_2!)
%t A342245 nn = 13; b[n_] := q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2;
%t A342245 e[x_] := Sum[x^n/b[n], {n, 0,nn}];Table[b[n],{n,0,nn}]CoefficientList[Series[e[x]^3, {x, 0, nn}], x]
%Y A342245 Cf. A132186, A005329.
%K A342245 nonn
%O A342245 0,2
%A A342245 _Geoffrey Critzer_, Mar 07 2021