A202784 -id:A202784 - cp's OEIS frontend

A067209 Number of n X n 0..1 matrices with all row and column sums equal.

1, 2, 4, 14, 140, 4322, 434542, 144109562, 165431317452, 654143160457922, 9331115832294448754, 469229841129645179004962, 87481268294773501231007850158, 58328998963405322376273800187396962, 147013017541698957281403494604070439260442

Offset: 0

R. H. Hardin, Feb 20 2002

nonn

			Some solutions for n=5
..1..1..0..0..1....0..0..1..0..1....0..1..0..1..1....1..1..0..1..0
..1..1..0..0..1....1..1..0..0..0....1..1..0..0..1....1..0..1..1..0
..0..0..1..1..1....1..0..0..1..0....1..1..1..0..0....1..0..1..0..1
..1..0..1..1..0....0..1..1..0..0....0..0..1..1..1....0..1..0..1..1
..0..1..1..1..0....0..0..0..1..1....1..0..1..1..0....0..1..1..0..1

Alois P. Heinz, Table of n, a(n) for n = 0..30
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums
B. D. McKay, 0-1 matrices with constant row and column sums

Row sums of A008300.

Column k=1 of A202784.

Added one term and example - R. H. Hardin, Jan 05 2012

a(0)=1 prepended and more terms (using data provided by B. D. McKay) from Alois P. Heinz, Apr 12 2017

A067213 Number of n X n 0..5 matrices with all row and column sums equal.

Original entry on oeis.org

1, 6, 36, 2442, 4756140, 255657603186, 394912380135575238, 18212451695911558370419098, 25800579898930843398657192741983084, 1146703506910134603350506378410721970397210150

Offset: 0

R. H. Hardin, Feb 20 2002

			Some solutions for n=3
..3..0..2....5..1..3....4..3..2....3..2..4....0..2..2....0..5..3....5..4..1
..2..0..3....1..4..4....0..4..5....5..4..0....0..2..2....3..3..2....2..3..5
..0..5..0....3..4..2....5..2..2....1..3..5....4..0..0....5..0..3....3..3..4

Column 5 of A202784.

a(0)=1 prepended and a(6)-a(7) from Andrew Howroyd, May 11 2020

a(8)-a(9) from Sean A. Irvine, Dec 13 2023

A067214 Number of n X n 0..6 matrices with all row and column sums equal.

Original entry on oeis.org

1, 7, 49, 5215, 21841937, 3430832678207, 21070560836988719941, 5258020825380244651432771257, 54855616345977485934984671841940089233

Offset: 0

R. H. Hardin, Feb 20 2002

			Some solutions for n=3
..4..0..6....5..4..4....1..2..4....5..5..2....5..0..4....6..3..0....2..6..2
..3..5..2....6..4..3....0..5..2....1..5..6....0..6..3....0..6..3....3..4..3
..3..5..2....2..5..6....6..0..1....6..2..4....4..3..2....3..0..6....5..0..5

Column 6 of A202784.

a(0)=1 prepended and a(6)-a(7) from Andrew Howroyd, May 11 2020

a(8) from Sean A. Irvine, Dec 12 2023

A067215 Number of n X n 0..7 matrices with all row and column sums equal.

Original entry on oeis.org

1, 8, 64, 10088, 82112704, 32701809796548, 664942846307414508064, 717494365786942428294915062416

Offset: 0

R. H. Hardin, Feb 20 2002

nonn

			Some solutions for n=3
..5..6..2....6..7..4....6..0..4....5..2..1....3..2..5....2..1..5....4..7..5
..3..4..6....6..4..7....0..4..6....3..4..1....4..1..5....5..2..1....7..2..7
..5..3..5....5..6..6....4..6..0....0..2..6....3..7..0....1..5..2....5..7..4

Column 7 of A202784.

a(0)=1 prepended and a(6)-a(7) from Andrew Howroyd, May 11 2020

A202785 Number of 3 X 3 0..n arrays with row and column sums equal.

Original entry on oeis.org

14, 87, 340, 1001, 2442, 5215, 10088, 18081, 30502, 48983, 75516, 112489, 162722, 229503, 316624, 428417, 569790, 746263, 964004, 1229865, 1551418, 1936991, 2395704, 2937505, 3573206, 4314519, 5174092, 6165545, 7303506, 8603647

Offset: 1

R. H. Hardin, Dec 24 2011

nonn

Row 3 of A202784.

			Some solutions for n=7:
..3..2..1....3..5..5....0..6..2....0..7..5....4..2..1....5..6..0....1..6..1
..2..0..4....5..6..2....2..1..5....6..1..5....3..2..2....0..4..7....5..2..1
..1..4..1....5..2..6....6..1..1....6..4..2....0..3..4....6..1..4....2..0..6

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Proof of empirical formula for A202785
Wikipedia, Ehrhart polynomial

Cf. A202784.

seq((3/10)*n^5 + (3/2)*n^4 + (7/2)*n^3 + (9/2)*n^2 + (16/5)*n + 1, n=1..30); # Robert Israel, May 02 2019

Empirical: a(n) = (3/10)*n^5 + (3/2)*n^4 + (7/2)*n^3 + (9/2)*n^2 + (16/5)*n + 1.
Conjectures from Colin Barker, Jun 01 2018: (Start)
G.f.: x*(7 - 2*x + x^2)*(2 + x + 4*x^2 - x^3) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)
Empirical formula verified (see link): Robert Israel, May 02 2019

A202786 Number of 4 X 4 0..n arrays with row and column sums equal.

Original entry on oeis.org

1, 140, 5673, 89520, 790425, 4756140, 21841937, 82112704, 264639729, 754898668, 1950230969, 4641494832, 10309971465, 21592075596, 42980713761, 81851507456, 149924818657, 265300850124, 455235310153, 759857498672, 1237071456633, 1968924291180, 3069774212913, 4696644466368

Offset: 0

R. H. Hardin, Dec 24 2011

nonn

From Robert Israel, May 03 2019: (Start)
a(n) is the number of integer lattice points in n*C where C is the polytope in R^(4 X 4) defined by Sum_{1<=i<=4} x_{i,j} = Sum_{1<=i<=4} x_{j,i} = Sum_{1<=i<=4} x_{i,1} for 1<=j<=4 and 0 <= x_{i,j} <= 1 for 1<=i,j<=4.
The vertices of this polytope have coordinates in {0,1/2,1} (an example of a vertex with non-integer coordinates is [0,1,1,1/2; 1,0,1,1/2; 1,1,0,1/2; 1/2,1/2,1/2,1]).
Therefore a(n) should be quasi-polynomial in n. (End)

			Some solutions for n=3
..2..2..1..3....2..1..2..1....3..2..1..0....1..3..2..2....0..3..2..2
..1..2..3..2....1..3..2..0....0..1..2..3....2..2..2..2....1..2..3..1
..3..1..3..1....2..1..1..2....1..1..1..3....2..1..2..3....3..2..1..1
..2..3..1..2....1..1..1..3....2..2..2..0....3..2..2..1....3..0..1..3

Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 1..21 from R. H. Hardin)
Wikipedia, Ehrhart quasi-polynomials

Row n=4 of A202784.

Conjecture: a(n) = (29 + 3*(-1)^n)/32 + (34/7)*n + (7202/525)*n^2 + (4658/189)*n^3 + (118873/3780)*n^4 + (5321/180)*n^5 + (36827/1800)*n^6 + (1285/126)*n^7 + (17581/5040)*n^8 + (2789/3780)*n^9 + (2789/37800)*n^10. - Robert Israel, May 03 2019

a(0)=1 prepended by Andrew Howroyd, Oct 14 2024

A067210 Number of n X n 0..2 matrices with all row and column sums equal.

Original entry on oeis.org

1, 3, 9, 87, 5673, 2577513, 8459809773, 207495141083975, 39125696231091082665, 57899300601363050181107733, 683382115404724657577266518831159, 65156055488094891194179319045960728827635, 50704966066306827486637758645370695372532720829085

Offset: 0

R. H. Hardin, Feb 20 2002

nonn

			Some solutions for n=4
..1..1..0..1....0..0..2..1....2..0..1..1....2..0..2..2....2..1..1..0
..0..2..1..0....1..1..1..0....0..1..2..1....1..2..1..2....0..2..0..2
..2..0..0..1....2..0..0..1....2..1..0..1....1..2..2..1....1..0..2..1
..0..0..2..1....0..2..0..1....0..2..1..1....2..2..1..1....1..1..1..1

Column k=2 of A202784.

a(7) and example from R. H. Hardin, Jan 05 2012
a(8)-a(12) from Andrew Howroyd, Jun 01 2017
a(0)=1 prepended by Andrew Howroyd, Oct 14 2024

A067211 Number of n X n 0..3 matrices with all row and column sums equal.

Original entry on oeis.org

1, 4, 16, 340, 89520, 290838284, 12092687195284, 6675338514190905402, 50327836869668591230004816, 5292118931181847999077738163123240, 7888144623867099277720906683959732404332016, 168815361056027801755263418243053345902341535059333028

Offset: 0

R. H. Hardin, Feb 20 2002

nonn

			Some solutions for n=3
..2..1..1....1..3..1....0..2..1....2..2..2....0..0..3....1..2..1....1..3..3
..0..1..3....1..1..3....3..0..0....2..2..2....2..1..0....0..1..3....3..1..3
..2..2..0....3..1..1....0..1..2....2..2..2....1..2..0....3..1..0....3..3..1

Column k=3 of A202784.

Added one term and example - R. H. Hardin, Jan 05 2012
a(7)-a(10) from Andrew Howroyd, Jun 01 2017
a(11) from Sean A. Irvine, Dec 14 2023
a(0)=1 prepended by Andrew Howroyd, Oct 14 2024

A067212 Number of n X n 0..4 matrices with all row and column sums equal.

Original entry on oeis.org

1, 5, 25, 1001, 790425, 11991502235, 3632894572698505, 22823286475857170293275, 3059096488092534208632573065881, 8933959513305390145234950731901862541009, 577809775169956916510410207413319463664222819419555

Offset: 0

R. H. Hardin, Feb 20 2002

nonn

			Some solutions for n=3
..4..4..3....2..1..1....2..1..1....4..0..4....1..2..4....3..2..2....3..2..1
..4..3..4....1..3..0....2..0..2....0..4..4....2..3..2....2..1..4....0..3..3
..3..4..4....1..0..3....0..3..1....4..4..0....4..2..1....2..4..1....3..1..2

Column k=4 of A202784.

Added one term and example - R. H. Hardin, Jan 05 2012
a(7)-a(8) from Andrew Howroyd, Jun 01 2017
a(9)-a(10) from Sean A. Irvine, Dec 14 2023
a(0)=1 prepended by Andrew Howroyd, Oct 14 2024

A067216 Number of n X n 0..8 matrices with all row and column sums equal.

Original entry on oeis.org

1, 9, 81, 18081, 264639729, 239657412990669, 14022194613733384575957, 55099298574634623527360069025765

Offset: 0

R. H. Hardin, Feb 20 2002

Column k=8 of A202784.

a(5)-a(7) from Sean A. Irvine, Dec 12 2023

a(0)=1 prepended by Andrew Howroyd, Oct 14 2024

cp's OEIS Frontend

A067209 Number of n X n 0..1 matrices with all row and column sums equal.

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A067213 Number of n X n 0..5 matrices with all row and column sums equal.

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A067214 Number of n X n 0..6 matrices with all row and column sums equal.

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A067215 Number of n X n 0..7 matrices with all row and column sums equal.

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A202785 Number of 3 X 3 0..n arrays with row and column sums equal.

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A202786 Number of 4 X 4 0..n arrays with row and column sums equal.

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A067210 Number of n X n 0..2 matrices with all row and column sums equal.

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A067211 Number of n X n 0..3 matrices with all row and column sums equal.

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A067212 Number of n X n 0..4 matrices with all row and column sums equal.

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A067216 Number of n X n 0..8 matrices with all row and column sums equal.

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Extensions