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, 34707333779115158227208335860718444216320, 332718225878012276874300952228513073208156487680

Offset: 0

Author

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

Keywords

• easy
• nonn

Examples

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

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.

Crossrefs

Cf. A002884.

Programs

Magma

[1] cat [(&*[2^n -2^k: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, May 16 2019

Mathematica

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

PARI

a(n) = prod(k=0, n-1, 2^n - 2^k)/n!; \\ Michel Marcus, Mar 25 2016

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

Formula

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

Extensions

More terms from Vladeta Jovovic, Apr 05 2000