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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.

%I A028362 #113 Apr 09 2025 15:01:44
%S A028362 1,3,15,135,2295,75735,4922775,635037975,163204759575,83724041661975,
%T A028362 85817142703524375,175839325399521444375,720413716161839357604375,
%U A028362 5902349576513949856852644375,96709997811181068404530578084375
%N A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.
%C A028362 These numbers appear in the second column of A155103. - _Mats Granvik_, Jan 20 2009
%C A028362 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
%D A028362 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630.
%H A028362 T. D. Noe, <a href="/A028362/b028362.txt">Table of n, a(n) for n = 1..50</a>
%H A028362 C. Bachoc and P. Gaborit, <a href="http://www.numdam.org/item?id=JTNB_2000__12_2_255_0">On extremal additive F_4 codes of length 10 to 18</a>, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
%H A028362 Steven T. Dougherty and Esengül Saltürk, <a href="https://doi.org/10.3934/amc.2024035">The neighbor graph of binary self-orthogonal codes</a>, Adv. Math. Comm. (2024). See p. 6.
%H A028362 Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.
%F A028362 a(n) = Product_{i=1..n-1} (2^i+1).
%F A028362 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.... = A079555 = Product_{k>=1} (1 + 1/2^k). - _Benoit Cloitre_, Jan 25 2003
%F A028362 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
%F A028362 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
%F A028362 From _Antti Karttunen_, Apr 15 2017: (Start)
%F A028362 a(n) = A048675(A285101(n-1)).
%F A028362 a(n) = b(n-1), where b(0) = 1, and for n > 0, b(n) = b(n-1) + (2^n)*b(n-1).
%F A028362 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].
%F A028362 (End)
%F A028362 G.f. A(x) satisfies: A(x) = x * (1 + A(2*x)) / (1 - x). - _Ilya Gutkovskiy_, Jun 06 2020
%F A028362 Conjectural o.g.f. as a continued fraction of Stieltjes type (S-fraction):
%F A028362   1/(1 - 3*x/(1 - 2*x/(1 - 10*x/(1 - 12*x/(1 - 36*x/(1 - 56*x/(1 - 136*x/(1 - 240*x/(1 - ... - 2^(n-1)*(2^n + 1)*x/(1 - 2^n*(2^n - 1)*x/(1 - ... ))))))))))). - _Peter Bala_, Sep 27 2023
%e A028362 G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ...
%p A028362 seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # _G. C. Greubel_, Jun 06 2020
%t A028362 Table[Product[2^i+1,{i,n-1}],{n,15}] (* or *) FoldList[Times,1, 2^Range[15]+1] (* _Harvey P. Dale_, Nov 21 2011 *)
%t A028362 Table[QPochhammer[-2, 2, n - 1], {n, 15}] (* _Arkadiusz Wesolowski_, Oct 29 2012 *)
%o A028362 (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
%o A028362 (PARI) {a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* _Michael Somos_, Jan 28 2018 */
%o A028362 (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 */
%o A028362 (Sage)
%o A028362 from ore_algebra import *
%o A028362 R.<x> = QQ['x']
%o A028362 A.<Qx> = OreAlgebra(R, 'Qx', q=2)
%o A028362 print((Qx - x - 1).to_list([0,1], 10))  # _Ralf Stephan_, Apr 24 2014
%o A028362 (Sage)
%o A028362 from sage.combinat.q_analogues import q_pochhammer
%o A028362 [q_pochhammer(n-1,-2,2) for n in (1..20)] # _G. C. Greubel_, Jun 06 2020
%o A028362 (Magma) [1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // _Vincenzo Librandi_, Dec 24 2015
%o A028362 (Python)
%o A028362 for n in range(2,40,2):
%o A028362   product = 1
%o A028362   for i in range(1,n//2-1 + 1):
%o A028362     product *= (2**i+1)
%o A028362   print(product)
%o A028362 # _Nathan J. Russell_, Mar 01 2016
%o A028362 (Python)
%o A028362 from math import prod
%o A028362 def A028362(n): return prod((1<<i)+1 for i in range(1,n)) # _Chai Wah Wu_, Jun 20 2022
%o A028362 (Scheme)
%o A028362 ;; With memoization-macro definec.
%o A028362 (define (A028362 n) (A028362off0 (- n 1)))
%o A028362 (definec (A028362off0 n) (if (zero? n) 1 (+ (A028362off0 (- n 1)) (* (expt 2 n) (A028362off0 (- n 1))))))
%o A028362 ;; _Antti Karttunen_, Apr 15 2017
%Y A028362 Cf. A000124, A003178, A003179, A028361, A028363, A048675, A053632, A068052 (XOR-analog), A285101.
%Y A028362 Cf. A155103. - _Mats Granvik_, Jan 20 2009
%Y A028362 Cf. A005329, A006088. - _Paul D. Hanna_, Sep 16 2009
%K A028362 nonn,easy,nice
%O A028362 1,2
%A A028362 _N. J. A. Sloane_