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A034348 Number of binary [ n,7 ] codes without 0 columns. 7
0, 0, 0, 0, 0, 0, 1, 7, 35, 170, 847, 4408, 24297, 143270, 901491, 5985278, 41175203, 287813284, 2009864185, 13848061942, 93369988436, 613030637339, 3908996099141, 24179747870890, 145056691643428, 844229016035010, 4769751989333029, 26181645303024760, 139750488576152520 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 05 2019

LINKS

Table of n, a(n) for n=1..29.

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=7.]

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,7,2}.]

Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]

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.

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 = 7 (this sequence) gives

print(A034253col(7, 30)) #

CROSSREFS

Cf. A034254, A034344, A034345, A034346, A034347, A034349, A253186.

Column k=7 of A034253 and first differences of A034361.

Sequence in context: A037099 A055421 A110213 * A249793 A268990 A005055

Adjacent sequences: A034345 A034346 A034347 * A034349 A034350 A034351

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Petros Hadjicostas, Oct 05 2019

STATUS

approved

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Last modified November 26 14:22 EST 2022. Contains 358362 sequences. (Running on oeis4.)