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A305737 Number of subsets S of vectors in GF(2)^n such that span(S) = GF(2)^n. 1

%I

%S 1,2,8,184,62464,4293001088,18446743803209556992,

%T 340282366920938461120638132973980614656,

%U 115792089237316195423570985008687907766497981100801256254562260326801824546816

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

%C Asymptotic to A001146(n) = 2^(2^n).

%D R. P. Stanley, Enumerative Combinatorics Vol 1, Cambridge, 1997, page 127.

%H Andrew Howroyd, <a href="/A305737/b305737.txt">Table of n, a(n) for n = 0..10</a>

%F a(n) = Sum_{k=0..n} A022166(n,k)*(-1)^(n-k)*2^binomial(n-k,2)*(2^(2^k)-1).

%t 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 (PARI) \\ here U(n,k) is A022166(n,k).

%o U(n,k)={polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}

%o 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 Cf. A001146, A022166.

%K nonn

%O 0,2

%A _Geoffrey Critzer_, Jun 22 2018

%E a(8) corrected by _Andrew Howroyd_, Mar 01 2020

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Last modified March 8 08:16 EST 2021. Contains 341942 sequences. (Running on oeis4.)