cp's OEIS Frontend

A289538 Expected dimension of the null space of a random linear operator on an n-dimensional vector space over the field with two elements as n -> infinity.

%I A289538 #22 Jul 17 2017 02:38:37
%S A289538 8,5,0,1,7,9,8,3,0,8,7,3,9,7,9,3,3,2,8,7,6,0,6,3,2,8,1,4,9,3,5,9,1,8,
%T A289538 7,8,8,4,0,4,2,6,7,2,5,9,7,3,2,0,2,7,2,5,9,8,7,3,5,8,0,5,2,5,5,6,3,0,
%U A289538 9,5,9,4,1,1,8,3,3,1,3,4,4,3,6,3,0,4,1,0,6,7,0,8,8,5,9,3,5,6,5,8
%N A289538 Expected dimension of the null space of a random linear operator on an n-dimensional vector space over the field with two elements as n -> infinity.
%C A289538 More precisely, let X:L(V) -> {0,1,2,...,n} be the random variable that assigns to each linear operator T on n-dimensional vector space V over F_2, the integer j in {0,1,2,...,n} such that the dimension of the null space of T = j. Then E(X) = 0.850179183...
%F A289538 Let A(x) = Sum_{n>=0} Product_{i>=n+1} (1-1/2^i)*x^n/A002884(n). Then A'(1) = 0.85017983...
%t A289538 nn = 300; q := 2;A[x_] := Sum[1/(FunctionExpand[QFactorial[j, q]] (q - 1)^j q^Binomial[j, 2]) Product[1 - 1/q^i, {i, j + 1, \[Infinity]}] x^j, {j, 0, nn}];RealDigits[
%t A289538   N[Normal[Series[D[A[x], x] /. x -> 1, {x, 0, nn}]], 100]][[1]]
%Y A289538 Cf. A002884, A182176, A289537, A289539, A289541, A289542.
%K A289538 cons,nonn
%O A289538 0,1
%A A289538 _Geoffrey Critzer_, Jul 10 2017