cp's OEIS Frontend

A305737 Number of subsets S of vectors in GF(2)^n such that span(S) = GF(2)^n.

%I A305737 #34 Apr 18 2025 11:29:19
%S A305737 1,2,8,184,62464,4293001088,18446743803209556992,
%T A305737 340282366920938461120638132973980614656,
%U A305737 115792089237316195423570985008687907766497981100801256254562260326801824546816
%N A305737 Number of subsets S of vectors in GF(2)^n such that span(S) = GF(2)^n.
%C A305737 Asymptotic to A001146(n) = 2^(2^n).
%D A305737 R. P. Stanley, Enumerative Combinatorics Vol 1, Cambridge, 1997, page 127.
%H A305737 Andrew Howroyd, <a href="/A305737/b305737.txt">Table of n, a(n) for n = 0..10</a>
%H A305737 Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Sequence_A305737">relationship to Boolean functions</a> (Wikiversity)
%F A305737 a(n) = Sum_{k=0..n} A022166(n,k)*(-1)^(n-k)*2^binomial(n-k,2)*(2^(2^k)-1).
%F A305737 Sum_{k=0..n} a(k)* A022166(n,k) = 2^(2^n) - 1. - _Geoffrey Critzer_, Apr 25 2024
%F A305737 a(n) = Sum_{k=0..n} A158474(n,k) * A051179(n-k). - _Tilman Piesk_, Mar 12 2025
%t A305737 Table[Sum[QBinomial[n, k, q] (-1)^(n - k) q^Binomial[n - k, 2] (2^(q^k) - 1) /. q -> 2, {k, 0, n}], {n, 0, 8}]
%o A305737 (PARI) \\ here U(n,k) is A022166(n,k).
%o A305737 U(n,k)={polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
%o A305737 a(n)={sum(k=0, n, U(n,k)*(-1)^(n-k)*2^binomial(n-k,2)*(2^(2^k)-1))} \\ _Andrew Howroyd_, Mar 01 2020
%Y A305737 Cf. A001146, A022166, A158474, A051179.
%K A305737 nonn
%O A305737 0,2
%A A305737 _Geoffrey Critzer_, Jun 22 2018
%E A305737 a(8) corrected by _Andrew Howroyd_, Mar 01 2020