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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. Théorie 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.<x> = QQ['x']; A.<Qx> = 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

N. J. A. Sloane

STATUS

approved

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Last modified June 19 05:22 EDT 2019. Contains 324217 sequences. (Running on oeis4.)