A008300 -id:A008300 - cp's OEIS frontend

A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 21, 24, 1, 1, 1, 5, 55, 282, 120, 1, 1, 1, 6, 120, 2008, 6210, 720, 1, 1, 1, 7, 231, 10147, 153040, 202410, 5040, 1, 1, 1, 8, 406, 40176, 2224955, 20933840, 9135630, 40320, 1, 1, 1, 9, 666, 132724, 22069251, 1047649905, 4662857360, 545007960, 362880, 1

Offset: 0

Alois P. Heinz, Apr 26 2015

Also the number of ordered factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity). A(2,2) = 3: (2*3)^2 = 36 = 4*9 = 6*6 = 9*4.

			Square array A(n,k) begins:
  1,   1,      1,        1,          1,           1,            1, ...
  1,   1,      1,        1,          1,           1,            1, ...
  1,   2,      3,        4,          5,           6,            7, ...
  1,   6,     21,       55,        120,         231,          406, ...
  1,  24,    282,     2008,      10147,       40176,       132724, ...
  1, 120,   6210,   153040,    2224955,    22069251,    164176640, ...
  1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Columns k=0-9 give: A000012, A000142, A000681, A001500, A172806, A172862, A172894, A172919, A172944, A172958.
Rows n=0+1, 2-9 give: A000012, A000027(k+1), A002817(k+1), A001496, A003438, A003439, A008552, A160318, A160319.
Main diagonal gives A110058.
Cf. A257463 (unordered factorizations), A333733 (non-isomorphic matrices), A008300 (binary matrices).

with(numtheory):
b:= proc(n, k) option remember; `if`(n=1, 1, add(
      `if`(bigomega(d)=k, b(n/d, k), 0), d=divisors(n)))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, k_] := b[n, k] = If[n==1, 1, Sum[If[PrimeOmega[d]==k, b[n/d, k], 0], {d, Divisors[n]}]]; A[n_, k_] := b[Product[Prime[i], {i, 1, n}]^k, k]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n, k)={
  local(M=Map(Mat([n, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
} \\ Andrew Howroyd, Apr 04 2020

bigomega = sloane.A001222
@cached_function
def b(n, k):
    if n == 1:
        return 1
    return sum(b(n//d, k) if bigomega(d) == k else 0 for d in n.divisors())
def A(n, k):
    return b(prod(nth_prime(i) for i in (1..n))^k, k)
[A(n, d-n) for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018, translated from Maple

from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([d-n]*n, [d-n]*n).cardinality() for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018

A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.

Original entry on oeis.org

1, 0, 1, 6, 90, 2040, 67950, 3110940, 187530840, 14398171200, 1371785398200, 158815387962000, 21959547410077200, 3574340599104475200, 676508133623135814000, 147320988741542099484000, 36574751938491748341360000, 10268902998771351157327104000

Offset: 0

N. J. A. Sloane

Or, number of labeled 2-regular relations of order n.

Also number of ways to arrange 2n rooks on an n X n chessboard, with no more than 2 rooks in each row and column (no 3 in a line). - Vaclav Kotesovec, Aug 03 2013

R. Bricard, L'Intermédiaire des Mathématiciens, 8 (1901), 312-313.
L. Comtet, Advanced Combinatorics, Reidel, 1974, Sect. 6.3 Multipermutations, pp. 235-236, P(n,2), bipermutations.
L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. Knuth. Reprinted in Problems in Applied Mathematics, ed. M. Klamkin, SIAM, 1990, p. 350.
Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
J. T. Lewis, Maximal L-free subsets of a squarefree array, Congressus Numerantium, 141 (1999), 151-155.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (b).
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992), pp. 152-153. [The second edition is said to be a better reference.]

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..48 from R. W. Robinson)
H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.
Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782.
Sally Cockburn and Joshua Lesperance, Deranged Socks, Mathematics Magazine, Vol. 86, no. 2, April 2013, pp. 97-109.
L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review, Vol. 21, No. 1 (Jan., 1979), p. 140. Solution by D. E. Knuth, SIAM Review, Vol. 22, No. 1 (Jan., 1980), pp. 101-102.
M. E. Kuczma, 0-1-Matrices with Line-Sums Equal to 2, Am. Math. Month. 99 (1992) 959-961, E3419.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Zhonghua Tan, Shanzhen Gao, and H. Niederhausen, Enumeration of (0,1) matrices with constant row and column sums, Appl. Math. Chin. Univ. 21 (4) (2006) 479-486.
Bo-Ying Wang and Fuzhen Zhang, On the precise number of (0,1)-matrices in A(R,S), Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From _N. J. A. Sloane_, Jun 07 2012
Index entries for sequences related to binary matrices

Cf. A000681, A053871, A123544 (connected relations), A000986 (symmetric matrices), A007107 (traceless matrices).
Cf. A000290, A002411, A005408.
Cf. A001501. Column 2 of A008300. Row sums of A284989.

a001499 n = a001499_list !! n
a001499_list = 1 : 0 : 1 : zipWith (*) (drop 2 a002411_list)
   (zipWith (+) (zipWith (*) [3, 5 ..] $ tail a001499_list)
                (zipWith (*) (tail a000290_list) a001499_list))
-- Reinhard Zumkeller, Jun 02 2013

a[n_] := (n-1)*n!*Gamma[n-1/2]*Hypergeometric1F1[2-n, 3/2-n, -1/2]/Sqrt[Pi]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 06 2011, after first formula *)

a(n)=if(n<2,n==0,(n^2-n)*(a(n-1)+(n-1)/2*a(n-2)))

seq(n)={Vec(serlaplace(serlaplace(exp(-x/2 + O(x*x^n))/sqrt(1-x + O(x*x^n)))))}; \\ Andrew Howroyd, Sep 09 2018

a(n) = (n! (n-1) Gamma(n-1/2) / Gamma(1/2) ) * 1F1[2-n; 3/2-n; -1/2] [Erlebach and Ruehr]. This representation is exact, asymptotic and convergent.
D-finite with recurrence 2*a(n) -2*n*(n-1)*a(n-1) -n*(n-1)^2*a(n-2)=0.
a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth]
a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand et al.)
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(-x/2)/sqrt(1-x); a(n) = n(n-1)/2 [ 2 a(n-1) + (n-1) a(n-2) ] (Bricard)
b_n = a_n/n! satisfies b_n = (n-1)(b_{n-1} + b_{n-2}/2); e.g.f. for {b_n} and for derangements (A000166) are related by D(x) = B(x)^2.
Limit_(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar, Sep 21 2007
a(n) = 4^(-n) * n!^2 * Sum_{i=0..n} (-2)^i * (2*n - 2*i)! / (i!*(n-i)!^2). - Shanzhen Gao, Feb 15 2010

A202784 T(n,k) is the number of n X n 0..k arrays with row and column sums equal.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 14, 1, 1, 5, 16, 87, 140, 1, 1, 6, 25, 340, 5673, 4322, 1, 1, 7, 36, 1001, 89520, 2577513, 434542, 1, 1, 8, 49, 2442, 790425, 290838284, 8459809773, 144109562, 1, 1, 9, 64, 5215, 4756140, 11991502235, 12092687195284, 207495141083975, 165431317452, 1

Offset: 0

R. H. Hardin, Dec 24 2011

			Table starts
n\k | 0            1               2              3                4
----+-------------------------------------------------------------------
  0 | 1            1               1              1                1 ...
  1 | 1            2               3              4                5 ...
  2 | 1            4               9             16               25 ...
  3 | 1           14              87            340             1001 ...
  4 | 1          140            5673          89520           790425 ...
  5 | 1         4322         2577513      290838284      11991502235 ...
  6 | 1       434542      8459809773 12092687195284 3632894572698505 ...
  7 | 1    144109562 207495141083975 ...
  8 | 1 165431317452 ...
  ...
Some solutions for n=4 k=3
..3..1..2..1....2..2..0..3....1..1..1..2....1..3..0..0....1..2..0..2
..2..3..2..0....1..2..2..2....3..0..1..1....1..0..1..2....3..1..0..1
..2..2..0..3....3..0..3..1....0..3..2..0....2..0..1..1....0..0..3..2
..0..1..3..3....1..3..2..1....1..1..1..2....0..1..2..1....1..2..2..0

Andrew Howroyd, Table of n, a(n) for n = 0..119 (first 15 antidiagonals; terms up to n=61 from R. H. Hardin)

Rows n=0..5 are A000012, A000027(k+1), A000290(k+1), A202785, A202786, A202787.
Columns k=1..15 are A067209, A067210, A067211, A067212, A067213, A067214, A067215, A067216, A067217, A067218, A067219, A067220, A067221, A067222, A067223.
Cf. A008300, A257493.

Added crossrefs. - R. H. Hardin, Jan 05 2012

Row and column zero inserted by Andrew Howroyd, Oct 14 2024

A001501 Number of n X n 0-1 matrices with all column and row sums equal to 3.

Original entry on oeis.org

1, 0, 0, 1, 24, 2040, 297200, 68938800, 24046189440, 12025780892160, 8302816499443200, 7673688777463632000, 9254768770160124288000, 14255616537578735986867200, 27537152449960680597739468800, 65662040698002721810659005184000

Offset: 0

N. J. A. Sloane

Also, for n >= 3, number of bicubical graphs on 2n labeled nodes of two colors [Read, 1958, 1971] - N. J. A. Sloane, Sep 08 2014

Also number of ways to arrange 3n rooks on an n X n chessboard, with no more than 3 rooks in each row and column (no 4 in a line). - Vaclav Kotesovec, Aug 03 2013

			G.f. = 1 + x^3 + 24*x^4 + 2040*x^5 + 297200*x^6 + 68938800*x^7 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,3).
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
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).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Example 1.1.3, page 2, f(n).
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.

Alois P. Heinz, Table of n, a(n) for n = 0..185 (first 51 terms from T. D. Noe)
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Bo-Ying Wang, Fuzhen Zhang, On the precise number of (0,1)-matrices in A(R,S), Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From _N. J. A. Sloane_, Jun 07 2012
Index entries for sequences related to binary matrices

Cf. A001499. Column 3 of A008300. Row sums of A284990.

a:= n-> n!^2/6^n *add(add((-1)^b *2^a *3^b *(3*n-3*a-2*b)!/
        (a! *b! *(n-a-b)!^2 *6^(n-a-b)), b=0..n-a), a=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 20 2011
# second Maple program:
a:= proc(n) option remember; `if`(n<4, (n-1)*(n-2)/2,
      n*(n-1)*(9*(3*n^2-5*n+4)*a(n-1)+(3*n-6)*(3*n+1)*
      (n-1)*a(n-2)+(9*n^2-30*n+13)*(n-1)*(n-2)^2*a(n-3)
      -(3*n-2)*(n-1)*(n-2)^2*(n-3)^2*a(n-4))/(36*n-60))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 13 2017

Table[6^(-n) Total[Map[(-1)^#[[2]] n!^2 (#[[2]] + 3 #[[3]])! 2^#[[1]] 3^#[[2]]/(#[[1]]! #[[2]]! #[[3]]!^2 6^#[[3]]) &, Compositions[n, 3]]], {n, 0, 20}] (* Geoffrey Critzer, Mar 19 2011 *)
a[n_] := n!^2*Sum[2^(2k-n)*3^(k-n)*(3(n-k))!*HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k )!^2), {k, 0, n}]/6^n;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 07 2018 *)

{a(n) = local(k); if( n<0, 0, n!^2 * sum(j=0, n, sum(i=0, n-j, if(1, k=n-i-j; (j + 3*k)! / (3^i * 36^k * i! * k!^2))) / (j! * (-2)^j)))}; /* Michael Somos, May 28 2002 */

a(n) = n!^2/6^n * Sum_{a=0..n} Sum_{b=0..n-a} (-1)^b * 2^a * 3^b * (3*n-3*a-2*b)! / (a! * b! * (n-a-b)!^2 * 6^(n-a-b)). - Shanzhen Gao, Feb 19 2010
D-finite with recurrence: 12*(3*n-5)*a(n) = 9*n*(3*n^2-5*n+4)*(n-1)*a(n-1) + 3*(n-2)*n*(3*n+1)*(n-1)^2*a(n-2) + (n-2)^2*n*(9*n^2-30*n+13)*(n-1)^2*a(n-3) - (n-3)^2*(n-2)^2*n*(3*n-2)*(n-1)^2*a(n-4). - Vaclav Kotesovec, Aug 03 2013
a(n) ~ sqrt(6*Pi) * (3/4)^n * n^(3*n+1/2) / exp(3*n+2). - Vaclav Kotesovec, Aug 03 2013

Additional comments from Michael Somos, May 28 2002

A133687 Triangle with number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to k (0<=k<=n), where equivalence is defined by row and column permutations.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 7, 4, 1, 1, 1, 1, 4, 16, 16, 4, 1, 1, 1, 1, 7, 51, 194, 51, 7, 1, 1, 1, 1, 8, 224, 3529, 3529, 224, 8, 1, 1, 1, 1, 12, 1165, 121790, 601055, 121790, 1165, 12, 1, 1, 1, 1, 14, 7454, 5582612, 156473848, 156473848, 5582612, 7454, 14, 1, 1

Offset: 0

Joost Vermeij (joost_vermeij(AT)live.nl), Jan 04 2008

T(n,k) = T(n,n-k). When 0 and 1 are switched, the number of equivalence classes remain the same.

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A008300. Burnside's lemma can be used to extend this method to the unlabeled case. This seems to require looping over partitions for both rows and columns. The number of partitions squared increases rapidly with n. For example, A000041(20)^2 = 393129. - Andrew Howroyd, Apr 03 2020

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1,   1;
  1, 1, 2,   1,    1;
  1, 1, 2,   2,    1,    1;
  1, 1, 4,   7,    4,    1,   1;
  1, 1, 4,  16,   16,    4,   1, 1;
  1, 1, 7,  51,  194,   51,   7, 1, 1;
  1, 1, 8, 224, 3529, 3529, 224, 8, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..189

Columns k=0..5 are A000012, A000012, A002865, A000512, A000513, A000516.
Row sums are A333681.
T(2n,n) gives A333740.
Cf. A000519, A008300 (labeled case), A008327 (bipartite graphs), A333159 (symmetric case).

Sum_{k=1..n} T(n, k) = A000519(n).

Missing a(72) inserted by Andrew Howroyd, Apr 01 2020

A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 31, 19, 1, 1, 1, 1, 51, 175, 175, 51, 1, 1, 1, 1, 141, 991, 2371, 991, 141, 1, 1, 1, 1, 393, 5881, 32611, 32611, 5881, 393, 1, 1, 1, 1, 1107, 35617, 481381, 1084851, 481381, 35617, 1107, 1, 1

Offset: 0

Andrew Howroyd, May 09 2020

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.

			Array begins:
====================================================================
n\k | 0 1   2     3       4          5            6            7
----|---------------------------------------------------------------
  0 | 1 1   1     1       1          1            1            1 ...
  1 | 1 1   1     1       1          1            1            1 ...
  2 | 1 1   3     7      19         51          141          393 ...
  3 | 1 1   7    31     175        991         5881        35617 ...
  4 | 1 1  19   175    2371      32611       481381      7343449 ...
  5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
  6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
  7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
     ...
The T(3,2) = 7 matrices are:
  [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
  [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
  [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]

Andrew Howroyd, Table of n, a(n) for n = 0..350 (first 26 antidiagonals)

Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
Main diagonal is A172645.
Cf. A008300, A333901, A376935, A377063 (up to row permutations).

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

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.

Original entry on oeis.org

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).

A321720 Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.

Original entry on oeis.org

1, 1, 2, 6, 25, 120, 726, 5040, 40410, 362881, 3630840, 39916800, 479069574, 6227020800, 87181402140, 1307674370040, 20922977418841, 355687428096000, 6402388104196400, 121645100408832000, 2432903379962038320, 51090942171778378800, 1124000886592995642000, 25852016738884976640000

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Andrew Howroyd, Table of n, a(n) for n = 0..180
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A057151, A068313, A008300, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321719, A321722, A321723.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(p) = p! for p prime as the squares are all permutation matrices of order p and a(n) >= n! for n > 1 (see comments in A321717 and A321719). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n, d<=n/d} A008300(n/d, d) for n > 0. - Andrew Howroyd, Apr 11 2020

a(7) from Chai Wah Wu, Jan 13 2019
a(8)-a(15) from Chai Wah Wu, Jan 14 2019
a(16)-a(21) from Chai Wah Wu, Jan 16 2019
Terms a(22) and beyond from Andrew Howroyd, Apr 11 2020

A058528 Number of n X n (0,1) matrices with all column and row sums equal to 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 120, 67950, 68938800, 116963796250, 315031400802720, 1289144584143523800, 7722015017013984456000, 65599839591251908982712750, 769237071909157579108571190000, 12163525741347497524178307740904300

Offset: 0

David desJardins, Dec 22 2000

nonn

Further terms generated by a Mathematica program written by Gordon G. Cash, who thanks B. R. Perez-Salvador, Universidad Autonoma Metropolitana Unidad Iztapalapa, Mexico, for providing the algorithm on which this program was based.
Also number of ways to arrange 4n rooks on an n X n chessboard, with no more than 4 rooks in each row and column. - Vaclav Kotesovec, Aug 04 2013
Generally (Canfield + McKay, 2004), a(n) ~ exp(-1/2) * binomial(n,s)^(2*n) / binomial(n^2,s*n), or a(n) ~ sqrt(2*Pi) * exp(-n*s-1/2*(s-1)^2) * (n*s)^(n*s+1/2) * (s!)^(-2*n). - Vaclav Kotesovec, Aug 04 2013

			a(4) = 1 because there is only one possible 4 X 4 (0,1) matrix with all row and column sums equal to 4, the matrix of all 1's. a(5) = 120 = 5! because there are 5X4X3X2X1 ways of placing a zero in each successive column (row) so that it is not in the same row (column) as any previously placed.

B. R. Perez-Salvador, S. de los Cobos Silva, M. A. Gutierrez-Andrade and A. Torres-Chazaro, A Reduced Formula for Precise Numbers of (0,1) Matrices in a(R,S), Disc. Math., 2002, 256, 361-372.

Vaclav Kotesovec, Table of n, a(n) for n = 0..150, [Computed with Maple program by Doron Zeilberger, see link below. This replaces an earlier b-file computed by Vladeta Jovovic (and corrected terms 26-31).]
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums
Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses, Maple package Bipartite; Local copy [Pdf file only, no active links]
B. D. McKay, 0-1 matrices with constant row and column sums
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Index entries for sequences related to binary matrices

Column 4 of A008300. Row sums of A284991.

a(n) = 24^{-n} sum_{alpha +beta + gamma + mu + u =n}frac{3^{ gamma }(-6)^{beta +u }8^{ mu }(n!)^{2}(4alpha +2 gamma + mu )!(beta +2 gamma )!}{alpha!beta! gamma! mu!u!} sum_{i=0}^{ floor (beta +2 gamma )/2 }frac{1}{24^{alpha - gamma +i}2^{beta +2 gamma -i}i!(beta +2 gamma -2i)!(alpha - gamma +i)!} - Shanzhen Gao, Nov 07 2007
From Vaclav Kotesovec, Aug 04 2013: (Start)
a(n) ~ exp(-1/2)*C(n,4)^(2*n)/C(n^2,4*n), (Canfield + McKay, 2004).
a(n) ~ sqrt(Pi)*2^(2*n+3/2)*9^(-n)*exp(-4*n-9/2)*n^(4*n+1/2).
(End)

More terms from Gordon G. Cash (cash.gordon(AT)epa.gov), Oct 22 2002

More terms from Vladeta Jovovic, Nov 12 2006

A260340 Triangle read by rows: T(n,k) = number of sets of linear n-ads in k variables.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 22, 22, 1, 1, 1, 1, 130, 550, 130, 1, 1, 1, 1, 822, 16700, 16700, 822, 1, 1, 1, 1, 6202, 703297, 3330915, 703297, 6202, 1, 1, 1, 1, 52552, 38135272, 957659906, 957659906, 38135272, 52552, 1, 1

Offset: 0

N. J. A. Sloane, Jul 30 2015

T(n,k) is the number of nonequivalent n X n binary matrices with k ones in every row and column up to permutation of rows. - Andrew Howroyd, Apr 18 2020

			Triangle begins:
  1;
  1, 1;
  1, 1,    1;
  1, 1,    1,      1;
  1, 1,    6,      1,       1;
  1, 1,   22,     22,       1,      1;
  1, 1,  130,    550,     130,      1,    1;
  1, 1,  822,  16700,   16700,    822,    1, 1;
  1, 1, 6202, 703297, 3330915, 703297, 6202, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows 0..20)
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.

Columns k=0..4 are A000012, A000012, A002137, A333899, A333900.
Row sums are A333891.
Cf. A008300, A133687, A257463.

T(n,k) = T(n,n-k). - Andrew Howroyd, Apr 18 2020

Extended to include k=0 and more terms added by Andrew Howroyd, Apr 18 2020

cp's OEIS Frontend

A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

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A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.

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A202784 T(n,k) is the number of n X n 0..k arrays with row and column sums equal.

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A001501 Number of n X n 0-1 matrices with all column and row sums equal to 3.

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A133687 Triangle with number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to k (0<=k<=n), where equivalence is defined by row and column permutations.

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A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

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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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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.