OFFSET
0,5
COMMENTS
"The familiar appearance of the first few rows [...] provides a good example of the perils of too hasty extrapolation in mathematics." - Slepian.
The difference between this triangle and the one for which it can be so easily mistaken is A250002. - Tilman Piesk, Nov 10 2014.
REFERENCES
M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994
LINKS
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [This is a rectangular array whose lower triangle contains T(n,k).]
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. [Apparently, the notation for T(n,k) is W_{nk2}; see p. 197.]
Petros Hadjicostas, Generating function for column k=4.
Petros Hadjicostas, Generating function for column k=5.
Petros Hadjicostas, Generating function for column k=6.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
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.
Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.]
FORMULA
From Petros Hadjicostas, Sep 30 2019: (Start)
T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.
G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).
G.f. for column k=3: -(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)^8).
G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. (See also some of the links above.)
(End)
EXAMPLE
k 0 1 2 3 4 5 6 7 8 9 10 11 sum
n
0 1 1
1 1 1 2
2 1 2 1 4
3 1 3 3 1 8
4 1 4 6 4 1 16
5 1 5 10 10 5 1 32
6 1 6 16 22 16 6 1 68
7 1 7 23 43 43 23 7 1 148
8 1 8 32 77 106 77 32 8 1 342
9 1 9 43 131 240 240 131 43 9 1 848
10 1 10 56 213 516 705 516 213 56 10 1 2297
11 1 11 71 333 1060 1988 1988 1060 333 71 11 1 6928
PROG
(Sage) # Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A076831col(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)/(1-x)
return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives
print(A076831col(4, 30)) # Petros Hadjicostas, Sep 30 2019
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Nov 21 2002
STATUS
approved