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A028365 Order of general affine group over GF(2), AGL(n,2). 4
1, 2, 24, 1344, 322560, 319979520, 1290157424640, 20972799094947840, 1369104324918194995200, 358201502736997192984166400, 375234700595146883504949480652800, 1573079924978208093254925489963584716800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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).

I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.

LINKS

Table of n, a(n) for n=0..11.

Putnam Exam. 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.

FORMULA

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

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

MAPLE

A028365 := n->2^n*product(2^n-2^'i', 'i'=0..n-1); # version 1

A028365 := n->product(2^'j'-1, 'j'=1..n)*2^binomial(n+1, 2); # version 2

MATHEMATICA

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 *)

(* Next, the connection with Vandermonde determinants *)

f[j_] := 2^(j - 1); z = 15;

v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]

Table[v[n], {n, 1, z}]   (* A203303 *)

Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A028365 *)

Table[v[n] v[n + 2]/(2*v[n + 1]^2), {n, 1, z - 1}]  (* A171499 *)

(* Clark Kimberling, Jan 01 2011 *)

PROG

(PARI) a(n)=if(n<0, 0, prod(k=1, n, 2^k-1)*2^((n^2+n)/2)) /* Michael Somos, May 09 2005 */

CROSSREFS

Cf. A020522.

Sequence in context: A213984 A137887 A232310 * A094050 A000479 A181231

Adjacent sequences:  A028362 A028363 A028364 * A028366 A028367 A028368

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 23 06:35 EDT 2014. Contains 247101 sequences.