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A132186 Number of idempotent n X n matrices over GF(2); also number of diagonalizable n X n matrices over GF(2).

%I A132186 #41 Jun 09 2025 07:38:29
%S A132186 1,2,8,58,802,20834,1051586,102233986,19614424834,7355623374338,
%T A132186 5494866505497602,8087844439442585602,23834930674299549249538,
%U A132186 138978138716920276085366786,1626809921636911219317749563394,37757678575184051755732304668884994
%N A132186 Number of idempotent n X n matrices over GF(2); also number of diagonalizable n X n matrices over GF(2).
%H A132186 Alois P. Heinz, <a href="/A132186/b132186.txt">Table of n, a(n) for n = 0..81</a>
%H A132186 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A132186 Kent E. Morrison, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F A132186 a(n) = sum(k=0...n, 2^(k(n-k))*[n,k]_2), where [n,k]_2 is the Gaussian binomial for q=2. - _Marc van Leeuwen_, May 22 2013
%F A132186 a(n)/A002884(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A002884(n))^2. - _Geoffrey Critzer_, Aug 04 2017
%F A132186 a(n) ~ c * 2^(n^2/2), where c = EllipticTheta(3, 0, 1/4) / QPochhammer(1/2) = 5.221199057419682876170323638731707664618893... if n is even and c = EllipticTheta(2, 0, 1/4) / QPochhammer(1/2) = 5.2043255482837364968664526298606149440286... if n is odd. - _Vaclav Kotesovec_, Jun 09 2025
%p A132186 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A132186       `if`(n=0, 1, T(n-1, k-1)+2^k*T(n-1, k)))
%p A132186     end:
%p A132186 a:= n-> add(2^(k*(n-k))*T(n, k), k=0...n):
%p A132186 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 06 2017
%t A132186 nn = 10; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
%t A132186 QFactorial[n, q]] /. q -> 2; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^2, {z, 0, nn}], z] (* _Geoffrey Critzer_, Aug 04 2017 *)
%t A132186 a[n_] := Block[{m}, Length@ Select[ Range[2^(n^2)], (m = Partition[ IntegerDigits[ #-1, 2, n^2], n]; Mod[m.m, 2] == m) &]]; a /@ Range[4] (* _Giovanni Resta_, Apr 09 2017 *)
%Y A132186 Cf. A002884, A081080.
%Y A132186 Row sums of A296548.
%K A132186 nonn
%O A132186 0,2
%A A132186 Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 19 2003 and _Vladeta Jovovic_, Nov 04 2007
%E A132186 This is the result of merging two independently submitted but identical sequences. Thanks to _Geoffrey Critzer_ for suggesting this. - _N. J. A. Sloane_, Dec 26 2017