The OEIS is supported by the many generous donors to the OEIS Foundation.



Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)



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


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.


(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)) #


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




N. J. A. Sloane.


More terms from Petros Hadjicostas, Oct 05 2019



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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