A053601 Number of bases of an n-dimensional vector space over GF(2).

1, 1, 3, 28, 840, 83328, 27998208, 32509919232, 132640470466560, 1927943976061501440, 100981078400558897823744, 19242660536873338307044442112, 13448310596010038676027219703234560

Offset

0,3

References

R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986

Links

G. C. Greubel, Table of n, a(n) for n = 0..59

Claude Carlet, Philippe Gaborit, Jon-Lark Kim and Patrick Sole, A new class of codes for Boolean masking of cryptographic computations, arXiv:1110.1193 [cs.IT], 2011-2012.

David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.

David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv:1604.01087 [quant-ph], 2016.

Formula

a(n) = (2^n-1)(2^n-2)...(2^n-2^(n-1))/n! = A002884(n)/n!.

Example

a(2)=3 because the 3 bases are {01,10}, {01,11}, {10,11}.

Mathematica

Table[Product[2^n - 2^k, {k, 0, n-1}]/n!, {n, 0, 20}] (* G. C. Greubel, May 16 2019 *)

Prog

(PARI) a(n) = prod(k=0, n-1, 2^n - 2^k)/n!; \\ Michel Marcus, Mar 25 2016
(Magma) [1] cat [(&*[2^n -2^k: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, May 16 2019
(Sage) [product(2^n -2^k for k in (0..n-1))/factorial(n) for n in (0..20)] # G. C. Greubel, May 16 2019

Crossrefs

Cf. A002884.

Sequence in context: A354664 A015474 A324462 * A328791 A140990 A196735

Adjacent sequences: A053598 A053599 A053600 * A053602 A053603 A053604

Keyword

easy,nonn

Author

Fred Galvin (galvin(AT)math.ukans.edu), Jan 20 2000

Extensions

More terms from Vladeta Jovovic, Apr 05 2000

Status

approved