A195644 -id:A195644 - cp's OEIS frontend

A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 20, 90, 20, 1, 1, 70, 1860, 1860, 70, 1, 1, 252, 44730, 297200, 44730, 252, 1, 1, 924, 1172556, 60871300, 60871300, 1172556, 924, 1, 1, 3432, 32496156, 14367744720, 116963796250, 14367744720, 32496156, 3432, 1, 1, 12870, 936369720, 3718394156400, 273957842462220, 273957842462220, 3718394156400, 936369720, 12870, 1

Offset: 0

Andrew Howroyd, Oct 11 2024

T(n,k) is the number of 2*n X 2*k {-1,1} matrices with all rows and columns summing to zero.

			Array begins:
========================================================================
n\k | 0   1       2           3               4                   5 ...
----+------------------------------------------------------------------
  0 | 1   1       1           1               1                   1 ...
  1 | 1   2       6          20              70                 252 ...
  2 | 1   6      90        1860           44730             1172556 ...
  3 | 1  20    1860      297200        60871300         14367744720 ...
  4 | 1  70   44730    60871300    116963796250     273957842462220 ...
  5 | 1 252 1172556 14367744720 273957842462220 6736218287430460752 ...
  ...

Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Main diagonal is A058527.
Columns 0..9 are A000012, A000984, A002896, A172556, A172555, A172557, A172558, A172559, A172560, A172554.
Cf. A008300, A195644, A333901, A334549, A377007 (up to permutations of rows and columns).

T(n, k)={
  local(M=Map(Mat([2*k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
  for(r=1, 2*n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
}

T(n,k) = T(k,n).

A195515 Number of lower triangles of an n X n -1..1 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 3, 15, 199, 6247, 505623, 105997283, 58923059879, 87882127554983, 355824020816195749, 3945677945542866439849, 120747614641791880577007593, 10263027079252827219477164325361, 2436143827877272419846561427146618539, 1622665432590314350761059585434700996826279

Offset: 1

R. H. Hardin, Sep 20 2011

nonn

			Some solutions for 4 X 4:
..0........0........0........0........0........0........0........0
..1-1.....-1.1......1-1......0.0.....-1.1......0.0.....-1.1......1-1
..0.0.0....1-1.0....0.0.0....0.0.0....0.0.0....1.0-1....0.0.0...-1.1.0
.-1.1.0.0..0.0.0.0.-1.1.0.0..0.0.0.0..1-1.0.0.-1.0.1.0..1-1.0.0..0.0.0.0

Column 1 of A195522 and column 2 of A195644.

a(13)-a(16) from Andrew Howroyd, May 16 2020

A195638 Number of lower triangles of an n X n 0..n array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 3, 19, 391, 25532, 5539434, 4116290300, 10699755803977, 98834805799836193, 3285452504354615242708, 397045217438108647483964418, 175907525023476787251523714204914, 287726899459800056272961335544498108972, 1747931723877860528675045077016711589788124993

Offset: 1

R. H. Hardin, Sep 21 2011

nonn

			Some solutions for n=4:
..1........1........1........1........1........1........1........1
..2.0......0.2......2.0......0.2......2.0......1.1......1.1......2.0
..1.1.1....1.1.1....0.2.1....2.1.0....1.2.0....0.2.1....2.1.0....1.0.2
..0.2.1.1..2.0.1.1..1.1.1.1..1.0.2.1..0.1.2.1..2.0.1.1..0.1.2.1..0.3.0.1

Diagonal of A195644.

a(12)-a(15) from Andrew Howroyd, May 16 2020

A195639 Number of lower triangles of an n X n 0..3 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 3, 19, 379, 22506, 4063437, 2303397986, 4165684145897, 24351094470461293, 465231630106636264261, 29310412705455253400706632, 6135104108066245513806738233622, 4293669524764808720927311131475805624, 10102086274414407316013385512463577654333437

Offset: 1

R. H. Hardin, Sep 21 2011

nonn

			Some solutions for n=4:
..1........1........1........1........1........1........1........1
..0.2......0.2......1.1......2.0......2.0......0.2......0.2......1.1
..0.1.2....3.0.0....2.0.1....1.2.0....1.0.2....2.1.0....3.0.0....2.0.1
..3.0.0.1..0.1.2.1..0.2.1.1..0.1.2.1..0.3.0.1..1.0.2.1..0.1.2.1..0.2.1.1

Column 3 of A195644.

a(12)-a(15) from Andrew Howroyd, May 16 2020

A195640 Number of lower triangles of an n X n 0..4 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 3, 19, 391, 25428, 5422820, 3868544673, 9374373474455, 78121590505558308, 2262375741577763676620, 229653423345084153764141412, 82310438395814327559991274396712, 104810153877236566878585186368604425875

Offset: 1

R. H. Hardin, Sep 21 2011

nonn

			Some solutions for n=4:
..1........1........1........1........1........1........1........1
..2.0......0.2......1.1......0.2......0.2......0.2......1.1......2.0
..1.0.2....1.0.2....1.0.2....1.0.2....3.0.0....3.0.0....1.0.2....1.0.2
..0.3.0.1..2.1.0.1..1.2.0.1..2.1.0.1..0.1.2.1..0.1.2.1..1.2.0.1..0.3.0.1

Column 4 of A195644.

a(12)-a(14) from Andrew Howroyd, May 16 2020

A195641 Number of lower triangles of an n X n 0..5 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 3, 19, 391, 25532, 5536654, 4102276124, 10576325574496, 96132191056438654, 3112190512707436846248, 361888107612909923618319047, 152218183506656720143679173709254, 233008581159686014846657554273887832751

Offset: 1

R. H. Hardin, Sep 21 2011

nonn

			Some solutions for n=5:
..1..........1..........1..........1..........1..........1..........1
..0.2........1.1........0.2........0.2........2.0........1.1........0.2
..0.1.2......0.1.2......0.2.1......2.1.0......2.1.0......0.1.2......1.0.2
..3.1.0.0....2.1.1.0....4.0.0.0....1.0.2.1....0.3.1.0....2.2.0.0....2.1.0.1
..1.0.1.2.1..1.1.0.2.1..0.0.2.2.1..1.1.1.1.1..0.0.2.2.1..1.0.1.2.1..1.1.1.1.1

Column 5 of A195644.

a(12)-a(14) from Andrew Howroyd, May 16 2020

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A376935 Array read by antidiagonals: T(n,k) is the number of 2*n X 2*k binary matrices with all row sums k and column sums n.

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A195515 Number of lower triangles of an n X n -1..1 array with all row and column sums zero.

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A195638 Number of lower triangles of an n X n 0..n array with all row sums equal to the length of the row and all column sums equal to the length of the column.

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A195639 Number of lower triangles of an n X n 0..3 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

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A195640 Number of lower triangles of an n X n 0..4 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

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A195641 Number of lower triangles of an n X n 0..5 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

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A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.