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A034344 Number of binary [ n,3 ] codes without 0 columns. 13
0, 0, 1, 3, 6, 12, 21, 34, 54, 82, 120, 174, 244, 337, 458, 613, 808, 1056, 1361, 1738, 2200, 2759, 3431, 4240, 5198, 6333, 7670, 9235, 11056, 13175, 15618, 18432, 21660, 25347, 29543, 34312, 39702, 45786, 52633, 60315, 68910, 78515, 89206, 101092, 114276, 128866, 144978, 162750, 182298 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The g.f. function below was calculated in Sage (using Fripertinger's method) and compared with the one in Lisonek's (2007) Example 5.3 (p. 627). - Petros Hadjicostas, Oct 02 2019

LINKS

Table of n, a(n) for n=1..49.

Discrete algorithms at the University of Bayreuth, Symmetrica.

Harald Fripertinger, Isometry Classes of Codes.

Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k = 3.]

H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,3,2}.]

Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5. The g.f. is given in Example 5.3.]

David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.

David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.

Wikipedia, Cycle index.

Wikipedia, Projective linear group.

FORMULA

G.f.: (x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^7) = (-x^15 + 2*x^14 - x^13 + x^12 + x^9 - x^7 + x^4 + x^3)/((1 - x)^2*(-x^2 + 1)*(-x^3 + 1)^2*(-x^4 + 1)*(-x^7 + 1)). - Petros Hadjicostas, Oct 02 2019

PROG

(Sage) # Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):

def A034253col(k, length):

    G1 = PSL(k, GF(2))

    G2 = PSL(k-1, GF(2))

    D1 = G1.cycle_index()

    D2 = G2.cycle_index()

    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)

    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)

    f = f1 - f2

    return f.taylor(x, 0, length).list()

# For instance the Taylor expansion for column k = 3 (this sequence) gives

print(A034253col(3, 30)) # Petros Hadjicostas, Oct 02 2019

CROSSREFS

Cf. A034254, A034345, A034346, A034347, A034348, A034349, A253186.

Column k=3 of A034253.

First differences of A034357.

Sequence in context: A011779 A161809 A084439 * A260640 A203292 A054578

Adjacent sequences:  A034341 A034342 A034343 * A034345 A034346 A034347

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Petros Hadjicostas, Oct 02 2019

STATUS

approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)