login
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
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
Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
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.
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
Column k=7 of A034253 and first differences of A034361.
Sequence in context: A055421 A373928 A110213 * A249793 A268990 A005055
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Petros Hadjicostas, Oct 05 2019
STATUS
approved