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A028362
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Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.
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12
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1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| These numbers appear in the second column of A155103. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 20 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2010: (Start)
a(n) = n terms in the sequence (1, 2, 4, 8, 16,...) dot n terms in the
sequence (1, 1, 3, 15, 135). Example: a(5) = 2295 = (1, 2, 4, 8, 16) dot
(1, 1, 3, 15, 135) = (1 + 2 + 12 + 120 + 2160). (End)
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REFERENCES
| C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. Theorie Nombres Bordeaux, 12 (2000), 255-271.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..50
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FORMULA
| Product( 2^i+1, i=1..n-1) (n>1).
Letting a(0)=1, we have a(n) = sum( k=0, n-1, 2^k*a(k) ) for n>0. a(n) is asymptotic to c*sqrt(2)^(n^2-n) where c=2.384231029031371724149899288....=prod(k>=1, 1+1/2^k). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 25 2003
G.f.: Sum_{n>=1} 2^(n*(n-1)/2) * x^n/[Product_{k=0..n-1} (1-2^k*x)]. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2009]
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MATHEMATICA
| Table[Product[2^i+1, {i, n-1}], {n, 15}] (* or *) FoldList[Times, 1, 2^Range[15]+1] (* From Harvey P. Dale, Nov 21 2011 *)
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PROG
| (PARI) {a(n)=polcoeff(sum(m=0, n, 2^(m*(m-1)/2)*x^m/prod(k=0, m-1, 1-2^k*x+x*O(x^n))), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2009]
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CROSSREFS
| Cf. A003178, A003179, A028363, A028361.
Cf. A155103. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 20 2009]
Cf. A006088, A005329. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2009]
Sequence in context: A006717 A059861 A030539 * A195764 A113723 A113379
Adjacent sequences: A028359 A028360 A028361 * A028363 A028364 A028365
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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