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 31-37 of 37 results.

A052367 Number of nonnegative integer 5 X 5 matrices with sum of elements equal to n, under row and column permutations.

Original entry on oeis.org

1, 1, 4, 10, 33, 91, 277, 792, 2341, 6654, 18802, 51508, 138147, 359457, 910756, 2240915, 5365106, 12495406, 28353714, 62725603, 135469991, 285904968, 590347527, 1193817552, 2366907846, 4605225266, 8801576140, 16538061290
Offset: 0

Views

Author

Vladeta Jovovic, Mar 08 2000

Keywords

Links

Crossrefs

Row 5 of A318795.
Cf. A002724, A053307, A052365, A052366.

Formula

G.f.: - (x^86 - 3*x^85 + 9*x^84 + 12*x^83 + 59*x^82 + 116*x^81 + 452*x^80 + 736*x^79 + 2080*x^78 + 3344*x^77 + 7312*x^76 + 11708*x^75 + 21793*x^74 + 32869*x^73 + 55563*x^72 + 79389*x^71 + 123072*x^70 + 168321*x^69 + 243961*x^68 + 319938*x^67 + 438431*x^66 + 553731*x^65 + 724251*x^64 + 885383*x^63 + 1111989*x^62 + 1318149*x^61 + 1600579*x^60 + 1845557*x^59 +
2172889*x^58 + 2444070*x^57 + 2798839*x^56 + 3076865*x^55 + 3436180*x^54 + 3696058*x^53 + 4034590*x^52 + 4250683*x^51 + 4541020*x^50 + 4689359*x^49 + 4909073*x^48 + 4972196*x^47 + 5102026*x^46 + 5069013*x^45 + 5102464*x^44 + 4971700*x^43 + 4909948*x^42 + 4688757*x^41 + 4542211*x^40 + 4249809*x^39 + 4036170*x^38 + 3694857*x^37 + 3438025*x^36 +
3075494*x^35 + 2800760*x^34 + 2442552*x^33 + 2174743*x^32 + 1843864*x^31 + 1602482*x^30 + 1316113*x^29 + 1114023*x^28 + 883313*x^27 + 725930*x^26 + 551915*x^25 + 439662*x^24 + 318308*x^23 + 245205*x^22 + 166823*x^21 + 124009*x^20 + 78506*x^19 + 56071*x^18 + 32361*x^17 + 22208*x^16 + 11357*x^15 + 7673*x^14 + 3221*x^13 + 2294*x^12 + 684*x^11 + 594*x^10 + 59*x^9 + 133*x^8 + 21*x^7 + 18*x^6 - 2*x^4 - 3*x^3 + 9*x^2 - 5*x + 1) divided by (see next line)
((x^20 - 1)*(x^11 - x^10 + x^6 - x^5 + x - 1)*(x^7 - 2*x^6 + x^5 + x^4 - x^3 - x^2 + 2*x - 1)*(x^4 + x^3 + x^2 + x + 1)^4*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^2 + 1)*(x^2 + 1)^5*(x^2 + x + 1)^5*(x + 1)^11*(x - 1)^22).

A052372 Number of nonnegative integer 6 X 6 matrices with sum of elements equal to n, under row and column permutations.

Original entry on oeis.org

1, 1, 4, 10, 33, 91, 298, 881, 2825, 8791, 27947, 87410, 272991, 837370, 2532012, 7496030, 21735743, 61570427, 170399621, 460413115, 1214983434, 3131870712, 7890604652, 19441462894, 46878788710, 110702854983, 256217556777
Offset: 0

Views

Author

Vladeta Jovovic, Mar 08 2000

Keywords

Links

Crossrefs

Row 6 of A318795.
Cf. A002724, A053307.

A052373 Number of nonnegative integer 7 X 7 matrices with sum of elements equal to n, under row and column permutations.

Original entry on oeis.org

1, 1, 4, 10, 33, 91, 298, 910, 2974, 9655, 32287, 108274, 367489, 1246921, 4229171, 14246120, 47542245, 156588539, 507914513, 1618965097, 5064384168, 15531406244, 46670874679, 137372332583, 396053582039, 1118577433593
Offset: 0

Views

Author

Vladeta Jovovic, Mar 08 2000

Keywords

Comments

a(n) = A007716(n) for n=0..7.

Links

Crossrefs

Row 7 of A318795.
Cf. A002724, A053307, A053304.

A363845 Triangle read by rows: T(n,k) = number of connected n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 6, 13, 6, 1, 0, 0, 1, 9, 34, 34, 9, 1, 0, 0, 1, 12, 76, 150, 76, 12, 1, 0, 0, 1, 16, 155, 558, 558, 155, 16, 1, 0, 0, 1, 20, 290, 1824, 3529, 1824, 290, 20, 1, 0, 0, 1, 25, 510, 5375, 19687, 19687, 5375, 510, 25, 1, 0, 0, 1, 30, 853, 14549, 98726, 194203, 98726, 14549, 853, 30, 1, 0
Offset: 0

Views

Author

Max Alekseyev, Jun 24 2023

Keywords

Comments

Also, rectangular array read by antidiagonals: A(m,n) = number of distinct m X n binary matrices M up to permutations of rows or columns such that M represents the reduced adjacency matrix of a connected bipartite graph.

Examples

			n=0: 1,
n=1: 1, 1,
n=2: 0, 1, 0,
n=3: 0, 1, 1, 0,
n=4: 0, 1, 2, 1, 0,
n=5: 0, 1, 4, 4, 1, 0,
n=6: 0, 1, 6, 13, 6, 1, 0,
n=7: 0, 1, 9, 34, 34, 9, 1, 0,
n=8: 0, 1, 12, 76, 150, 76, 12, 1, 0,
n=9: 0, 1, 16, 155, 558, 558, 155, 16, 1, 0,
...

Links

Crossrefs

Inverse bivariate Euler transform of A028657.
Cf. A002724, A363846.

A116976 Number of nonsingular n X n matrices with rational entries equal to 0 or 1, up to row and column permutations.

Original entry on oeis.org

1, 2, 8, 61, 1153, 64310, 11352457, 6417769762
Offset: 1

Views

Author

Vladeta Jovovic, Apr 01 2006

Keywords

Comments

"Rational entries" means that a matrix is nonsingular iff it has a nonzero determinant. (Over the integers a matrix with determinant > 1 is not invertible.) M. F. Hasler, May 25 2020

Examples

			From _M. F. Hasler_, May 25 2020: (Start)
Representatives of the two inequivalent nonsingular (0,1) matrices for n=2 are
  [ 1  0 ]   and   [ 1  1 ]  .
  [ 0  1 ]         [ 0  1 ]
For n=3 we have 8 nonsingular nonequivalent representatives:
  [1 0 0]  [1 0 0]  [1 0 1]  [1 1 1]  [1 1 0]  [1 1 0]  [1 1 1]  [1 1 0]
  [0 1 0], [0 1 1], [0 1 1], [0 1 0], [0 1 1], [1 0 1], [0 1 1], [1 0 1].
  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 1]  [0 0 1]  [1 1 1]
To see that they are inequivalent, consider their column sums:
  (1 1 1), (1 1 2), (1 1 3), (1 2 2), (1 2 2), (2 2 2), (1 2 3), (3 2 2).
Only the 4th and 5th matrix have equivalent column sum signature (1,2,2), but their row sums are (3,1,1) resp. (2,2,1). Therefore they can't be obtained one from the other by row and column permutations which leave invariant these sums.
(End)

Links

Crossrefs

Cf. A055165, A002724.

Formula

a(n) = A002724(n) - A116977(n). - Max Alekseyev, Jul 14 2022

Extensions

a(8) from Brendan McKay, May 25 2020

A116977 Number of n X n rational {0,1}-matrices of determinant 0, up to row and column permutations.

Original entry on oeis.org

1, 5, 28, 256, 4471, 187300, 22290203, 8267860926
Offset: 1

Views

Author

Vladeta Jovovic, Apr 01 2006

Keywords

Links

Crossrefs

Cf. A046747, A002724, A116976.

Formula

a(n) = A002724(n) - A116976(n). - Max Alekseyev, Feb 28 2010

Extensions

a(8) from Alois P. Heinz, Jun 30 2022

A177793 Partial sums of A054247.

Original entry on oeis.org

1, 3, 9, 111, 8659, 4220403, 8594777715, 70377477369459, 2305913405481561715, 302233760834929839713907, 158456627262298939528655810163, 332307157402856267706609817833582195
Offset: 0

Views

Author

Jonathan Vos Post, May 13 2010

Keywords

Comments

Partial sums of number of n X n binary matrices under action of dihedral group of the square D_4. Can this ever be prime?

Examples

			a(4) = 1 + 2 + 6 + 102 + 8548 = 8659 = 7 * 1237.

Crossrefs

Cf. A002724, A054247, A054407.

Programs

  • PARI
    A054247(n)={ local(a) ; if(n%2==0, a=2^(n^2)+2*2^(n^2/4)+3*2^(n^2/2)+2*2^((n^2+n)/2), a=2^(n^2)+2*2^((n^2+3)/4)+2^((n^2+1)/2)+4*2^((n^2+n)/2); ) ; return(a/8) ; }
A177793(n)={ return(sum(i=0,n,A054247(i))) ; }
{ for(n=0,20, print1(A177793(n),",") ; ) ; } (End)

Formula

a(n) = SUM[i=0..n] A054247(i) = SUM[i=0..n] [(1/8)*(2^(i^2)+2*2^(i^2/4)+3*2^(i^2/2)+2*2^((i^2+i)/2)) if i is even and (1/8)*(2^(i^2)+2*2^((i^2+3)/4)+2^((i^2+1)/2)+4*2^((i^2+i)/2)) if i is odd].

Extensions

Extended by R. J. Mathar, May 28 2010
Previous Showing 31-37 of 37 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.