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.

Previous Showing 11-18 of 18 results.

A002725 Number of incidence matrices: n X (n+1) binary matrices under row and column permutations.

Original entry on oeis.org

1, 3, 13, 87, 1053, 28576, 2141733, 508147108, 402135275365, 1073376057490373, 9700385489355970183, 298434346895322960005291, 31479360095907908092817694945, 11474377948948020660089085281068730, 14568098446466140788730090352230460100956
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(0) = 1 by convention.

Examples

			a(1) = 3: [0,0], [0,1], [1,1].
a(2) = 13:
000 000 000 000 001 001 001 001 001 011 011 011 111
000 001 011 111 001 010 011 110 111 011 101 111 111

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002623, A002727, A006148, A002728, A002724.
A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*
      coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/
      mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/
      mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),
      t=b(n+1$2)), s=b(n$2)):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 01 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
a[n_] := Sum[Sum[2^Sum[ Sum [ GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[ i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+1, n+1]}], {s,  b[n, n]}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)
  • PARI
    a(n) = A(n+1,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

Formula

a(n) = sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+1} (fix A[s_1, s_2, ...; t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fix A[...] = 2^sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Sean A. Irvine, Jul 31 2014

Extensions

More terms from Vladeta Jovovic, Feb 04 2000

A002728 Number of n X (n+2) binary matrices.

Original entry on oeis.org

1, 4, 22, 190, 3250, 136758, 17256831, 7216495370, 10271202313659, 49856692830176512, 826297617412284162618, 46948445432190686211183650, 9200267975562856184153936960940, 6261904454889790650636380541051266410, 14910331834338546882501064075429145637985605
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002623, A002727, A006148, A002724, A002725.
A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*
      coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/
      mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/
      mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),
      t=b(n+2$2)), s=b(n$2)):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 01 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Table[Function[{p}, p + j*x^i]@ b[n-i*j, i-1] , {j, 0, n/i}]]] // Flatten; a[n_] := Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+2, n+2]}], {s, b[n, n]}]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 28 2014, after Alois P. Heinz *)
  • PARI
    a(n) = A(n+2,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

Formula

a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+2} (fix A[s_1, s_2, ...;t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fix A[...] = 2^sum {i, j>=1} (gcd(i, j)*s_i*t_j). - Sean A. Irvine, Jul 31 2014

Extensions

More terms from Vladeta Jovovic, Feb 04 2000

A005746 Number of n-covers of an unlabeled 4-set.

Original entry on oeis.org

1, 9, 51, 230, 863, 2864, 8609, 23883, 61883, 151214, 350929, 778113, 1656265, 3398229, 6743791, 12983181, 24311044, 44377016, 79124476, 138048542, 236050912, 396137492, 653285736, 1059923072, 1693592112, 2667563553, 4145373780, 6360553548, 9643151582
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Number of n X 4 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

A diagonal of A055080.
First differences give A055082.
Cf. A002727, A006148.
Cf. A005744, A005745, A005747, A005748, A005771.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x (1 + 3 x + 9 x^2 + 26 x^3 + 35 x^4 + 92 x^5 + 127 x^6 + 201 x^7 + 242 x^8 + 253 x^9 + 248 x^10 + 205 x^11 + 123 x^12 + 86 x^13 + 31 x^14 + 24 x^15 + 19 x^16 + 5 x^17 + 3 x^18 -
2 x^19 - 4 x^20 + 2 x^21 - 4 x^22 + 3 x^23 - x^25 + 2 x^26 - x^27)/((1 - x)^16 (1 + x)^6 (1 + x^2)^3 (1 + x + x^2)^4), {x, 0, 29}], x] (* Michael De Vlieger, Aug 23 2016 *)
  • PARI
    Vec(x*(1 +3*x +9*x^2 +26*x^3 +35*x^4 +92*x^5 +127*x^6 +201*x^7 +242*x^8 +253*x^9 +248*x^10 +205*x^11 +123*x^12 +86*x^13 +31*x^14 +24*x^15 +19*x^16 +5*x^17 +3*x^18 -2*x^19 -4*x^20 +2*x^21 -4*x^22 +3*x^23 -x^25 +2*x^26 -x^27) / ((1 -x)^16*(1 +x)^6*(1 +x^2)^3*(1 +x +x^2)^4) + O(x^40)) \\ Colin Barker, Aug 23 2016
  • PARI
    Vec(G(4, x) - G(3, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

Formula

a(n) = A006148(n) - A002727(n).
G.f.: x*(1 +3*x +9*x^2 +26*x^3 +35*x^4 +92*x^5 +127*x^6 +201*x^7 +242*x^8 +253*x^9 +248*x^10 +205*x^11 +123*x^12 +86*x^13 +31*x^14 +24*x^15 +19*x^16 +5*x^17 +3*x^18 -2*x^19 -4*x^20 +2*x^21 -4*x^22 +3*x^23 -x^25 +2*x^26 -x^27) / ((1 -x)^16*(1 +x)^6*(1 +x^2)^3*(1 +x +x^2)^4). - Corrected by Colin Barker, Aug 23 2016

Extensions

More terms and g.f. from Vladeta Jovovic, May 26 2000
a(19) onwards corrected by Sean A. Irvine, Aug 22 2016

A005771 Number of n-covers of an unlabeled 5-set.

Original entry on oeis.org

1, 12, 103, 736, 4571, 25326, 127415, 588687, 2518997, 10053739, 37656707, 133084998, 445949359, 1422934989, 4340110439, 12697803333, 35744330644, 97081519369, 255032046536, 649459943602, 1606518048420, 3867119228081, 9073566868140, 20783186834063
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Number of n X 5 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

References

  • R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

A diagonal of A055080.
First differences give A055083.
Cf. A006148, A052264.

Programs

Formula

a(n) = A052264(n) - A006148(n). - Andrew Howroyd, Feb 28 2023

Extensions

More terms from Vladeta Jovovic, Jun 13 2000
Terms a(21) and beyond from Andrew Howroyd, Feb 28 2023

A353585 Square array T(n,k): row n lists the number of inequivalent matrices over Z/nZ, modulo permutations of rows and columns, of size r X c, 1 <= r <= c, c >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 7, 6, 4, 1, 4, 27, 10, 5, 1, 13, 10, 76, 15, 6, 1, 36, 92, 20, 175, 21, 7, 1, 5, 738, 430, 35, 351, 28, 8, 1, 22, 15, 8240, 1505, 56, 637, 36, 9, 1, 87, 267, 35, 57675, 4291, 84, 1072, 45, 10, 1, 317, 5053, 1996, 70, 289716, 10528, 120, 1701, 55, 11
Offset: 1

Views

Author

M. F. Hasler, Apr 28 2022

Keywords

Comments

The array is read by falling antidiagonals.
Each row lists the number of inequivalent matrices of size 1 X 1, then 2 X 1, 2 X 2, then 3 X 1, 3 X 2, 3 X 3, etc., with coefficients in Z/nZ (or equivalently, in {1, ..., n}). See Examples for more.
Row 1 counts the zero matrices, there is only one of any size. Row 2 counts binary matrices, this is the lower triangular part of A028657, without the trivial row & column 0. (This table might have been extended with a trivial column 0 = A000012 (counting the 1 matrix of size 0) and row 0 = A000007 counting the number of r X c matrices with no entry, as done in A246106.)
The square matrices (size 1 X 1, 2 X 2, 3 X 3, ...) are counted in columns with triangular numbers, k = T(r) = r(r+1)/2 = (1, 3, 6, 10, 15, ...) = A000217.

Examples

			The table starts
   n \ k=1,  2,   3,   4,   5,   6, ...: T(n,k)
  ----+--------------------------------------
   1  |  1   1    1    1    1     1 ...
   2  |  2   3    7    4   13    36 ...
   3  |  3   6   27   10   92   738 ...
   4  |  4  10   76   20  430  8240 ...
   5  |  5  15  175   35 1505 57675 ...
  ...
Columns 2, 3 and 4, 5, 6 correspond to matrices of size 1 X 2, 2 X 2 and 1 X 3, 2 X 3, 3 X 3, respectively.
Column 4 says that there are (1, 4, 10, 20, 35, ...) inequivalent matrices of size 1 X 3 with entries in Z/nZ (n = 1, 2, 3, 4, ...); these numbers are given by (n+2 choose 3) = binomial(n+2, 3) = n(n+1)(n+2)/6 = A000292(n).

Links

Crossrefs

All of the following related sequences can be expressed in terms of T(n, k, r) := T(n, k(k-1)/2 + r), WLOG r <= k:
A028657(n,k) = A353585(2,n,k): inequivalent m X n binary matrices,
A002723(n) = T(2,n,2): size n X 2, A002724(n) = T(2,n,n): size n X n,
A002727(n) = T(2,n,3): size n X 3, A002725(n) = T(2,n,n+1): size n X (n+1),
A006148(n) = T(2,n,4): size n X 4, A002728(n) = T(2,n,n+2): size n X (n+2),
A052264(n) = T(2,n,5): size n X 5,
A052269(n) = T(3,n,n): number of inequivalent ternary matrices of size n X n,
A052271(n) = T(4,n,n): number of inequivalent matrices over Z/4Z of size n X n,
A052272(n) = T(5,n,n): number of inequivalent matrices over Z/5Z of size n X n,
A246106(n,k) = A353585(k,n,n): number of inequivalent n X n matrices over Z/kZ, and its diagonal A091058 and columns 1, 2, ..., 10: A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.

Programs

  • PARI
    A353585(n,k,r)={if(!r,r=sqrtint(8*k)\/2; k-=r*(r-1)\2); my(m(c, p=1, L=0)=for(i=1,#c, if(i==#c || c[i+1]!=c[i], p *= c[i]^(i-L)*(i-L)!; L=i )); p, S=0); forpart(P=k, my(T=0); forpart(Q=r, T += n^sum(i=1,#P, sum(j=1,#Q, gcd(P[i],Q[j]) ))/m(Q)); S += T/m(P)); S}

Formula

Let k = c(c-1)/2 + r, 1 <= r <= c, then
T(n, c, r) := T(n, k) = Sum_{p in P(c), q in P(r)} n^S(p, q)/(N(p)*N(q)), where P(r) are the partitions of r, S(p, q) = Sum_{i in p, j in q} gcd(i, j), N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p.
(See, e.g., A080577 for a list of partitions of positive integers.)
In particular:
T(n, 1) = n, T(n, 2) = n(n+1)/2 = A000217(n), T(n, 4) = C(n+2, 3) = A000292(n), T(n, 7) = C(n+3, 4) = A000332(n+3), etc.: T(n, k(k+1)/2 + 1) = C(n+k, k+1),
T(n, k(k+1)/2) = A246106(k, n).

A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 6, 16, 81, 299, 1358, 5567, 23350, 91998, 351058, 1269907, 4394634, 14495236, 45779246, 138567568, 403282017, 1130773069, 3062535192, 8028046724, 20411824364, 50429813556, 121280243676, 284360432241, 650972702410
Offset: 0

Views

Author

Vladeta Jovovic, Aug 05 2000

Keywords

References

  • M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.

Links

Crossrefs

Column k=5 of A363349.
Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

Formula

G.f.: 1/3840*(1/(1 - x^1)^32 + 231/(1 - x^2)^16 + 20/(1 - x^1)^16/(1 - x^2)^8 + 520/(1 - x^4)^8 + 60/(1 - x^1)^8/(1 - x^2)^12 + 80/(1 - x^1)^8/(1 - x^3)^8 + 720/(1 - x^2)^4/(1 - x^6)^4 + 160/(1 - x^1)^4/(1 - x^2)^2/(1 - x^3)^4/(1 - x^6)^2 + 320/(1 - x^4)^2/(1 - x^12)^2 + 240/(1 - x^1)^4/(1 - x^2)^2/(1 - x^4)^6 + 480/(1 - x^8)^4 + 240/(1 - x^2)^4/(1 - x^4)^6 + 384/(1 - x^1)^2/(1 - x^5)^6 + 384/(1 - x^2)^1/(1 - x^10)^3).

A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 7, 23, 153, 849, 6128, 43534, 319119, 2255466, 15307395, 98349144, 597543497, 3430839916, 18653684881, 96273409815, 473010823993, 2218614773950, 9961651259869, 42927432229913, 177963663264430
Offset: 0

Views

Author

Vladeta Jovovic, Aug 05 2000

Keywords

References

  • M. A. Harrison, On the number of classes of binary matrices, IEEE Trans.Computers, 22 (1973), 1048-1051.

Links

Crossrefs

Column k=6 of A363349.
Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

Formula

G.f.: 1/46080*(1/(1 - x^1)^64 + 1053/(1 - x^2)^32 + 30/(1 - x^1)^32/(1 - x^2)^16 + 4920/(1 - x^4)^16 + 180/(1 - x^1)^16/(1 - x^2)^24 + 120/(1 - x^1)^8/(1 - x^2)^28 + 160/(1 - x^1)^16/(1 - x^3)^16 + 5280/(1 - x^2)^8/(1 - x^6)^8 + 960/(1 - x^1)^8/(1 - x^2)^4/(1 - x^3)^8/(1 - x^6)^4 + 3840/(1 - x^4)^4/(1 - x^12)^4 + 640/(1 - x^1)^4/(1 - x^3)^20 + 1920/(1 - x^2)^2/(1 - x^6)^10 + 720/(1 - x^1)^8/(1 - x^2)^4/(1 - x^4)^12 + 5760/(1 - x^8)^8 + 2160/(1 - x^2)^8/(1 - x^4)^12 + 1440/(1 - x^1)^4/(1 - x^2)^6/(1 - x^4)^12 + 2304/(1 - x^1)^4/(1 - x^5)^12 + 6912/(1 - x^2)^2/(1 - x^10)^6 + 3840/(1 - x^1)^2/(1 - x^2)^1/(1 - x^3)^2/(1 - x^6)^9 + 3840/(1 - x^4)^1/(1 - x^12)^5).

A055537 Number of asymmetric types of (4,n)-hypergraphs under action of symmetric group S_4.

Original entry on oeis.org

7, 64, 352, 1485, 5245, 16290, 45830, 119042, 289367, 664878, 1455136, 3051762, 6163153, 12033700, 22792660, 41997387, 75463460, 132510654, 227803866, 384031745, 635752338, 1034842530, 1658130458, 2617965384, 4076707044
Offset: 3

Views

Author

Vladeta Jovovic, Jul 09 2000

Keywords

Examples

			There are 7 asymmetric (4,3)-hypergraphs: {{1,2},{1,2,3},{1,3,4}}, {{1,2},{1,3},{1,2,4}}, {{1,2},{1,2},{1,3}}, {{1},{1,2},{1,2,3}}, {{1},{2,3},{1,2,4}}, {{1},{1,2},{2,3}}, {{1},{2},{1,3}}.

Crossrefs

Cf. A006148.

Formula

G.f.: (1/(1-x)^16-6/(1-x)^8/(1-x^2)^4-3/(1-x)^4/(1-x^2)^6+8/(1-x)^4/(1-x^3)^4+2/(1-x)^2/(1-x^2)^3/(1-x^4)^2+6/(1-x)^4/(1-x^2)^4/(1-x^4)-8/(1-x)^2/(1-x^4)^2/(1-x^6))/24

Extensions

More terms from James Sellers, Jul 11 2000
Previous Showing 11-18 of 18 results.

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

Content is available under The OEIS End-User License Agreement.