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 A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n. 26
 1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These numbers appear in the second column of A155103. - Mats Granvik, Jan 20 2009 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). - Gary W. Adamson, Aug 02 2010 REFERENCES F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630. LINKS T. D. Noe, Table of n, a(n) for n=1..50 C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. ThÃ©orie Nombres Bordeaux, 12 (2000), 255-271. FORMULA a(n) = Product_{i=1..n-1} (2^i+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.... = Product_{k>=1} (1 + 1/2^k). - Benoit Cloitre, Jan 25 2003 G.f.: Sum_{n>=1} 2^(n*(n-1)/2) * x^n/(Product_{k=0..n-1} (1-2^k*x)). - Paul D. Hanna, Sep 16 2009 a(n) = 2^(binomial(n,2) - 1)*(-1; 1/2)_{n}, where (a;q)_{n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015 From Antti Karttunen, Apr 15 2017: (Start) a(n) = A048675(A285101(n-1)). a(n) = b(n-1), where b(0) = 1, and for n > 0, b(n) = b(n-1) + (2^n)*b(n-1). a(n) = Sum_{i=1..A000124(n-1)} A053632(n-1,i-1)*(2^(i-1)). [Where the indexing of both rows and columns of irregular table A053632(row,col) is considered to start from zero.] (End) EXAMPLE G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ... MATHEMATICA Table[Product[2^i+1, {i, n-1}], {n, 15}] (* or *) FoldList[Times, 1, 2^Range[15]+1] (* Harvey P. Dale, Nov 21 2011 *) Table[QPochhammer[-2, 2, n - 1], {n, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *) 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)} \\ Paul D. Hanna, Sep 16 2009 (PARI) {a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* Michael Somos, Jan 28 2018 */ (PARI) {a(n) = sum(k=0, n-1, 2^(k*(k+1)/2) * prod(j=1, k, (2^(n-j) - 1) / (2^j - 1)))}; /* Michael Somos, Jan 28 2018 */ (Sage) from ore_algebra import * R. = QQ['x'] A. = OreAlgebra(R, 'Qx', q=2) print((Qx - x - 1).to_list([0, 1], 10))  # Ralf Stephan, Apr 24 2014 (MAGMA) [1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015 (PYTHON) for n in range(2, 50, 2):   product = 1   for i in range(1, n/2-1 + 1):     product *= (2**i+1) print product # Nathan J. Russell, Mar 01 2016 (Scheme, with memoization-macro definec) (define (A028362 n) (A028362off0 (- n 1))) (definec (A028362off0 n) (if (zero? n) 1 (+ (A028362off0 (- n 1)) (* (expt 2 n) (A028362off0 (- n 1)))))) ;; Antti Karttunen, Apr 15 2017 CROSSREFS Cf. A000124, A003178, A003179, A028363, A028361, A048675, A053632, A068052 (XOR-analog), A285101. Cf. A155103. - Mats Granvik, Jan 20 2009 Cf. A006088, A005329. - Paul D. Hanna, Sep 16 2009 Sequence in context: A059861 A232699 A030539 * A195764 A113723 A113379 Adjacent sequences:  A028359 A028360 A028361 * A028363 A028364 A028365 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified March 29 03:09 EDT 2020. Contains 333104 sequences. (Running on oeis4.)