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A028365
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Order of general affine group over GF(2), AGL(n,2).
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4
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1, 2, 24, 1344, 322560, 319979520, 1290157424640, 20972799094947840, 1369104324918194995200, 358201502736997192984166400, 375234700595146883504949480652800, 1573079924978208093254925489963584716800
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OFFSET
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0,2
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COMMENTS
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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. [From Clark Kimberling, Jan 01 2012]
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REFERENCES
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Putnam Exam. 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
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.
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LINKS
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Table of n, a(n) for n=0..11.
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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}] (* From 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 *)
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PROG
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(PARI) a(n)=if(n<0, 0, prod(k=1, n, 2^k-1)*2^((n^2+n)/2)) /* Michael Somos May 09 2005 */
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CROSSREFS
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Cf. A020522.
Sequence in context: A136524 A213984 A137887 * A094050 A000479 A181231
Adjacent sequences: A028362 A028363 A028364 * A028366 A028367 A028368
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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