cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).

Original entry on oeis.org

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
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			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;
  ...

Links

Crossrefs

This is A076831 with the k=0 column omitted.
Columns include A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).
Cf. A034253, A034254, A034328.

Programs

  • 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
print(A034356col(4, 30)) # Petros Hadjicostas, Oct 07 2019

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=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)

A076831 Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 16, 22, 16, 6, 1, 1, 7, 23, 43, 43, 23, 7, 1, 1, 8, 32, 77, 106, 77, 32, 8, 1, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 1, 11, 71, 333, 1060, 1988, 1988
Offset: 0

Views

Author

N. J. A. Sloane, Nov 21 2002

Keywords

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.

Examples

			     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

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

Crossrefs

Cf. A006116, A022166, A076766 (row sums).
A034356 gives same table but with the k=0 column omitted.
Columns include A000012 (k=0), A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).

Programs

  • 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

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)

A034348 Number of binary [ n,7 ] codes without 0 columns.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

Crossrefs

Cf. A034254, A034344, A034345, A034346, A034347, A034349, A253186.
Column k=7 of A034253 and first differences of A034361.

Programs

  • 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)) #

Extensions

More terms from Petros Hadjicostas, Oct 05 2019
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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