A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375

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. Théorie Nombres Bordeaux, 12 (2000), 255-271.

Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

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.... = A079555 = 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)

G.f. A(x) satisfies: A(x) = x * (1 + A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020

Conjectural o.g.f. as a continued fraction of Stieltjes type (S-fraction):

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

Example

G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ...

Maple

seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # G. C. Greubel, Jun 06 2020

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.<x> = QQ['x']
A.<Qx> = OreAlgebra(R, 'Qx', q=2)
print((Qx - x - 1).to_list([0, 1], 10))  # Ralf Stephan, Apr 24 2014
(Sage)
from sage.combinat.q_analogues import q_pochhammer
[q_pochhammer(n-1, -2, 2) for n in (1..20)] # G. C. Greubel, Jun 06 2020
(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, 40, 2):
  product = 1
  for i in range(1, n//2-1 + 1):
    product *= (2**i+1)
  print(product)
# Nathan J. Russell, Mar 01 2016
(Python)
from math import prod
def A028362(n): return prod((1<<i)+1 for i in range(1, n)) # Chai Wah Wu, Jun 20 2022
(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, A028361, A028363, A048675, A053632, A068052 (XOR-analog), A285101.

Cf. A155103. - Mats Granvik, Jan 20 2009

Cf. A005329, A006088. - Paul D. Hanna, Sep 16 2009

Sequence in context: A348420 A232699 A030539 * A195764 A113723 A113379

Adjacent sequences: A028359 A028360 A028361 * A028363 A028364 A028365

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved