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 1-10 of 12 results. Next

A034188 Number of binary codes of length 3 with n words.

Original entry on oeis.org

1, 1, 3, 3, 6, 3, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.

Links

Crossrefs

Cf. A034189, A034190, A034191, A034192, A034193, A034194, A034195, A034196, A034197.
A row of A039754.

A039754 Irregular triangle read by rows: T(n,k) = number of binary codes of length n with k words (n >= 0, 0 <= k <= 2^n); also number of 0/1-polytopes with vertices from the unit n-cube; also number of inequivalent Boolean functions of n variables with exactly k nonzero values under action of Jevons group.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 6, 3, 3, 1, 1, 1, 1, 4, 6, 19, 27, 50, 56, 74, 56, 50, 27, 19, 6, 4, 1, 1, 1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, 158658, 133576, 98804, 65664, 38073, 19963, 9013, 3779, 1326, 472, 131, 47, 10, 5, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For N=1 through N=5, the first 2^(N-1) terms of row N are also found in triangle A171871, which is related to A005646. This was shown for all N by Andrew Weimholt, Dec 30 2009. [Robert Munafo, Jan 25 2010]

Examples

			Triangle begins:
  k  0  1  2  3   4   5   6   7   8   9  10  11  12 13 14 15 16   sums
n
0    1  1                                                            2
1    1  1  1                                                         3
2    1  1  2  1   1                                                  6
3    1  1  3  3   6   3   3   1   1                                 22
4    1  1  4  6  19  27  50  56  74  56  50  27  19  6  4  1  1    402

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 112.
  • M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 150.

Links

Crossrefs

Row sums give A000616. Cf. A052265.
Rows give A034188, A034189, A034190, etc.
Columns are A034198, A034199, A034200, etc.
Diagonal is A276412.
For other versions of this triangle see A171876, A039754, A276777.
Cf. A171871.

Programs

  • Mathematica
    P = IntegerPartitions;
AC[d_Integer] := Module[{C, M, p},(* from W. Y. C. Chen algorithm *) M[p_List] := Plus @@ p!/(Times @@ p * Times @@ (Length /@ Split[p]!)); C[p_List, q_List] := Module[{r, m, k, x}, r = If[0 == Length[q], 1, 2*2^IntegerExponent[LCM @@ q, 2]]; m = LCM @@ Join[p/GCD[r, p], q/GCD[r, q]]; CoefficientList[Expand[Product[(1 + x^(k *r))^((Plus @@ Map[MoebiusMu[k/#]*2^Plus @@ GCD[#*r, Join[p, q]]&, Divisors[k]])/(k*r)), {k, 1, m}]], x]]; Sum[Binomial[d, p]*Plus @@ Plus @@ Outer[M[#1] M[#2] C[#1, #2]*2^(d - Length[#1] - Length[#2]) &, P[p], P[d - p], 1], {p, 0, d}]/(d! 2^d)]; AC[0]  = {1, 1};
AC /@ Range[0, 5] // Flatten (* Jean-François Alcover, Dec 15 2019, after Robert A. Russell in A034189 *)
Table[ CoefficientList[ CycleIndexPolynomial[ GraphData[ {"Hypercube", n}, "AutomorphismGroup"], Array[Subscript[x, ##] &, 2^n]] /. Table[ Subscript[x, i] -> 1 + x^i, {i, 1, 2^n}], x], {n, 1,8}] // Grid (* Geoffrey Critzer, Jan 10 2020 *)

Formula

Reference gives g.f.
Fripertinger gives g.f. for the number of classes of (n, m) nonlinear codes over an alphabet of size A.

Extensions

Corrected and extended by Vladeta Jovovic, Apr 20 2000
Entry revised by N. J. A. Sloane, Sep 19 2016
T(0, 1) = 1 inserted. (There are two 0-ary functions.) - Tilman Piesk, Jan 10 2023

A034190 Number of binary codes of length 5 with n words.

Original entry on oeis.org

1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, 158658, 133576, 98804, 65664, 38073, 19963, 9013, 3779, 1326, 472, 131, 47, 10, 5, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of 2-colorings of the vertices of the 5-cube having n nodes of one color.

References

  • W. Y. C. Chen, Induced cycle structures of the hyperoctahedral group. SIAM J. Disc. Math. 6 (1993), 353-362.
  • H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.
  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.

Links

Crossrefs

Row n=5 of A039754.
Cf. A034188, A034189, A034191, A034192, A034193, A034194, A034195, A034196, A034197.
Cf. A171872 and A171876. - Robert Munafo, Jan 25 2010

Programs

  • Mathematica
    (* From Robert A. Russell, May 08 2007: (Start) *)
P[ n_Integer ]:=P[ n ]=P[ n,n ];P[ n_Integer,_ ]:={}/;(n<0);(* partitions *)
P[ 0, ]:={{}};P[ n_Integer,1 ]:={Table[ 1,{n} ]};P[ ,0 ]:={};(*S.S. Skiena*)
P[ n_Integer,m_Integer ]:=Join[ Map[ (Prepend[ #,m ])&,P[ n-m,m ] ],P[ n,m-1 ] ];
AC[ d_Integer ]:=Module[ {C,M,p}, (* from W.Y.C. Chen algorithm *)
M[ p_List ]:=Plus@@p!/(Times@@p Times@@(Length/@Split[ p ]!));
C[ p_List,q_List ]:=Module[ {r,m,k,x},r=If[ 0==Length[ q ],1,2 2^
IntegerExponent[ LCM@@q,2 ] ];m=LCM@@Join[ p/GCD[ r,p ],q/GCD[ r,q ] ];
CoefficientList[ Expand[ Product[ (1+x^(k r))^((Plus@@Map[ MoebiusMu[ k/# ]
2^Plus@@GCD[# r,Join[ p,q ] ]&,Divisors[ k ] ])/(k r)),{k,1,m} ] ],x ] ];
Sum[ Binomial[ d,p ]Plus@@Plus@@Outer[ M[ #1 ]M[ #2 ]C[ #1,#2 ]2^(d-Length[ #1 ]-Length[ #2 ])&,P[ p ],P[ d-p ],1 ],{p,0,d} ]/(d!2^d) ];AC[ 5 ]
(* End *)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 11 2007

A034191 Number of binary codes of length 6 with n words.

Original entry on oeis.org

1, 1, 6, 16, 103, 497, 3253, 19735, 120843, 681474, 3561696, 16938566, 73500514, 290751447, 1052201890, 3492397119, 10666911842, 30064448972, 78409442414, 189678764492, 426539774378, 893346071377, 1745593733454
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of 2-colorings of the vertices of the 6-cube having n nodes of one color.
The b-file shows the full sequence.

References

  • W. Y. C. Chen, Induced cycle structures of the hyperoctahedral group. SIAM J. Disc. Math. 6 (1993), 353-362.
  • H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.
  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.

Links

Crossrefs

Row n=6 of A039754.
Cf. A034188, A034189, A034190, A034192, A034193, A034194, A034195, A034196, A034197.

Programs

  • Mathematica
    (* From Robert A. Russell, May 08 2007: (Start) *)
P[ n_Integer ]:=P[ n ]=P[ n,n ];P[ n_Integer,_ ]:={}/;(n<0);(* partitions *)
P[ 0, ]:={{}};P[ n_Integer,1 ]:={Table[ 1,{n} ]};P[ ,0 ]:={};(*S.S. Skiena*)
P[ n_Integer,m_Integer ]:=Join[ Map[ (Prepend[ #,m ])&,P[ n-m,m ] ],P[ n,m-1 ] ];
AC[ d_Integer ]:=Module[ {C,M,p}, (* from W.Y.C. Chen algorithm *)
M[ p_List ]:=Plus@@p!/(Times@@p Times@@(Length/@Split[ p ]!));
C[ p_List,q_List ]:=Module[ {r,m,k,x},r=If[ 0==Length[ q ],1,2 2^
IntegerExponent[ LCM@@q,2 ] ];m=LCM@@Join[ p/GCD[ r,p ],q/GCD[ r,q ] ];
CoefficientList[ Expand[ Product[ (1+x^(k r))^((Plus@@Map[ MoebiusMu[ k/# ]
2^Plus@@GCD[# r,Join[ p,q ] ]&,Divisors[ k ] ])/(k r)),{k,1,m} ] ],x ] ];
Sum[ Binomial[ d,p ]Plus@@Plus@@Outer[ M[ #1 ]M[ #2 ]C[ #1,#2 ]2^(d-Length[ #1 ]-Length[ #2 ])&,P[ p ],P[ d-p ],1 ],{p,0,d} ]/(d!2^d) ];AC[ 6 ]
(* End *)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 11 2007

A034192 Number of binary codes of length 7 with n words.

Original entry on oeis.org

1, 1, 7, 23, 203, 1606, 18435, 221778, 2773763, 33297380, 375158732, 3907656327, 37504171766, 331785257145, 2712509085687, 20560611034067, 144992583036707, 954428916508309, 5882732966056385, 34048050206744705
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Row n=7 of A039754.
Cf. A034188, A034189, A034190, A034191, A034193, A034194, A034195, A034196, A034197.

A034193 Number of binary codes of length 8 with n words.

Original entry on oeis.org

1, 1, 8, 32, 373, 4647, 91028, 2074059, 51107344, 1245930065, 28900653074, 625715497344, 12562875567065, 233750783834504, 4038807303045625, 65003434860142353, 977872935273906860, 13795944871933252078
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Row n=8 of A039754.
Cf. A034188, A034189, A034190, A034191, A034192, A034194, A034195, A034196, A034197.

A034194 Number of binary codes of length 9 with n words.

Original entry on oeis.org

1, 1, 9, 43, 649, 12320, 404154, 16957301, 805174011, 38921113842, 1816451773537, 79799396735243, 3267743403989063, 124448560749072651, 4413401558241969897, 146147123072487323183, 4533679418476771721737
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Row n=9 of A039754.
Cf. A034188, A034189, A034190, A034191, A034192, A034193, A034195, A034196, A034197.

A034195 Number of binary codes of length 10 with n words.

Original entry on oeis.org

1, 1, 10, 56, 1079, 30493, 1646000, 124727148, 11244522420, 1063289204514, 98630203059528, 8687099045170277, 716661691995997667, 55145074470990131334, 3958951571937696919325
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A034188, A034189, A034190, A034191, A034192, A034193, A034194, A034196, A034197.

A034196 Number of binary codes of length 11 with n words.

Original entry on oeis.org

1, 1, 11, 71, 1727, 71218, 6232542, 840130123, 142255064793, 26089682147964, 4776270743160543, 838032011887365999, 138409456269288832810, 21380210213948881081642, 3086290263475796210940632
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A034188, A034189, A034190, A034191, A034192, A034193, A034194, A034195, A034197.

A034197 Number of binary codes of length 12 with n words.

Original entry on oeis.org

1, 1, 12, 89, 2681, 158374, 22151853, 5244256701, 1653293013642, 584494025830394, 210237300968913438, 73212582945836596465, 24126483152959778734211, 7456837714994642537616273
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Last term is a(2^12) = 1. - Sean A. Irvine, Aug 07 2020

Links

Crossrefs

Cf. A034188, A034189, A034190, A034191, A034192, A034193, A034194, A034195, A034196.
Showing 1-10 of 12 results. Next

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

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