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A034356
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Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).
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23
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1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 16, 22, 16, 6, 1, 7, 23, 43, 43, 23, 7, 1, 8, 32, 77, 106, 77, 32, 8, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 11, 71, 333, 1060, 1988, 1988, 1060, 333, 71, 11, 1, 12, 89
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OFFSET
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1,2
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LINKS
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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. [We have T(n,k) = W_{nk2}; see p. 197.]
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FORMULA
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T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.
G.f. for column k=1: x/(1-x)^2.
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. For some cases, see also the links above.
(End)
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EXAMPLE
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Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
1;
2, 1;
3, 3, 1;
4, 6, 4, 1;
5, 10, 10, 5, 1;
6, 16, 22, 16, 6, 1;
7, 23, 43, 43, 23, 7, 1;
8, 32, 77, 106, 77, 32, 8, 1;
...
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PROG
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(Sage) # Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A034356col(k, length):
R = PowerSeriesRing(ZZ, 'x', default_prec=length)
x = R.gen().O(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.list()
# For instance the Taylor expansion for column k = 4 gives
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CROSSREFS
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This is A076831 with the k=0 column omitted.
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KEYWORD
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AUTHOR
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STATUS
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approved
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