cp's OEIS Frontend

A028365 Order of general affine group over GF(2), AGL(n,2).

%I A028365 #57 Jun 05 2025 10:04:22
%S A028365 1,2,24,1344,322560,319979520,1290157424640,20972799094947840,
%T A028365 1369104324918194995200,358201502736997192984166400,
%U A028365 375234700595146883504949480652800,1573079924978208093254925489963584716800
%N A028365 Order of general affine group over GF(2), AGL(n,2).
%C A028365 For n > 0, a(n) = v(n+1)/v(n), where v = A203305 is the Vandermonde determinant of the first n of the numbers -2^j - 1; see the Mathematica section. - _Clark Kimberling_, Jan 01 2012
%D A028365 J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 54 (1.64).
%H A028365 Seiichi Manyama, <a href="/A028365/b028365.txt">Table of n, a(n) for n = 0..57</a>
%H A028365 Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, <a href="https://arxiv.org/abs/2505.20434">SZ Sequences: Binary-Based (0, 2^q)-Sequences</a>, arXiv:2505.20434 [cs.GR], 2025. See p. 7.
%H A028365 Marcus Brinkmann, <a href="https://www.researchgate.net/publication/332564840_Extended_Affine_and_CCZ_Equivalence_Classes_up_to_Dimension_4">Extended Affine and CCZ Equivalence up to Dimension 4</a>, Ruhr University Bochum (2019).
%H A028365 Putnam Competition 1999, <a href="http://math.ucsd.edu/~pfitz/downloads/putnam/putnam1999.pdf">Question A6</a>, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
%H A028365 I. Strazdins, <a href="https://doi.org/10.1023/A:1005769927571">Universal affine classification of Boolean functions</a>, Acta Applic. Math. 46 (1997), 147-167.
%F A028365 a(n) is asymptotic to C*2^(n*(n+1)) where C = 0.288788095086602421278899721... = prod(k>=1, 1-1/2^k) (cf. A048651). - _Benoit Cloitre_, Apr 11 2003
%F A028365 a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)). [From Putman Exam]. - _Max Alekseyev_, May 18 2007
%F A028365 a(n) = 2*A203305(n), n > 0. - _Clark Kimberling_, Jan 01 2012
%F A028365 From _Max Alekseyev_, Jun 09 2015: (Start)
%F A028365 a(n) = 2^A000217(n) * A005329(n).
%F A028365 a(n) = 2^n * A002884(n).
%F A028365 a(n) = 2^n * n! * A053601(n). (End)
%F A028365 From _G. C. Greubel_, Aug 31 2023: (Start)
%F A028365 a(n) = Product_{j=0..n-1} (2^(n+1) - 2^(j+1)).
%F A028365 a(n) = (-1)^n * 2^binomial(n+1,2) * QPochhammer(2,2,n). (End)
%p A028365 A028365 := n->2^n*product(2^n-2^'i','i'=0..n-1); # version 1
%p A028365 A028365 := n->product(2^'j'-1,'j'=1..n)*2^binomial(n+1,2); # version 2
%t A028365 RecurrenceTable[{a[1]==1, a[2]==2, a[3]==24, a[n]==(6a[n-1]^2 a[n-3] - 8a[n-1] a[n-2]^2)/(a[n-2] a[n-3])}, a[n], {n,20}] (* _Harvey P. Dale_, Aug 03 2011 *)
%t A028365 (* Next, the connection with Vandermonde determinants *)
%t A028365 f[j_]:= 2^j - 1; z = 15;
%t A028365 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A028365 Table[v[n], {n,z}]   (* A203303 *)
%t A028365 Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
%t A028365 Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *) (* _Clark Kimberling_, Jan 01 2011 *)
%t A028365 Table[(-1)^n*2^Binomial[n+1,2]*QPochhammer[2,2,n], {n,0,20}] (* _G. C. Greubel_, Aug 31 2023 *)
%o A028365 (PARI) a(n)=if(n<0,0,prod(k=1,n,2^k-1)*2^((n^2+n)/2)) /* _Michael Somos_, May 09 2005 */
%o A028365 (Magma) [1] cat [(&*[2^(n+1) - 2^(j+1): j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 31 2023
%o A028365 (SageMath) [product(2^(n+1) - 2^(k+1) for k in range(n)) for n in range(21)] # _G. C. Greubel_, Aug 31 2023
%Y A028365 Cf. A000217, A002884, A005329, A020522, A048651, A053601, A203305.
%K A028365 nonn,easy,nice
%O A028365 0,2
%A A028365 _N. J. A. Sloane_