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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
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,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.
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
Column k=3 of A034253.
First differences of A034357.
Sequence in context: A011779 A161809 A084439 * A260640 A203292 A054578
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Petros Hadjicostas, Oct 02 2019
STATUS
approved

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Last modified July 25 01:41 EDT 2024. Contains 374585 sequences. (Running on oeis4.)