A284990 -id:A284990 - cp's OEIS frontend

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

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

A284989 Triangle T(n,k) read by rows: the number of n X n {0,1} matrices with trace k where each row sum and each column sum is 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 3, 2, 9, 24, 24, 24, 9, 216, 540, 610, 420, 210, 44, 7570, 18000, 20175, 13720, 6300, 1920, 265, 357435, 829920, 909741, 617610, 284235, 91140, 19005, 1854, 22040361, 50223600, 54295528, 36663312, 17072790, 5679184, 1337280, 203952, 14833

Offset: 0

R. J. Mathar, Apr 07 2017

			0:         1
1:         0        0
2:         0        0        1
3:         1        0        3        2
4:         9       24       24       24        9
5:       216      540      610      420      210      44
6:      7570    18000    20175    13720     6300    1920     265
7:    357435   829920   909741   617610   284235   91140   19005   1854
8:  22040361 50223600 54295528 36663312 17072790 5679184 1337280 203952 14833

Alois P. Heinz, Rows n = 0..14, flattened
Index entries for sequences related to binary matrices

Cf. A001499 (row sums), A000166 (diagonal), A007107 (column 0).

Cf. A008290, A098825, A284990, A284991.

P(n, t='t) = {
  my(z=vector(n, k, eval(Str("z", k))),
     s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
     f=vector(n, k, t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
  for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
  for (k=1, n, g=polcoef(g, 2, z[k]));
  g;
};
seq(N) = concat([[1], [0, 0], [0, 0, 1]], apply(n->Vec(P(n)), [3..N]));
concat(seq(8)) \\ Gheorghe Coserea, Dec 21 2018

Let z1..zn be n variables and s1 = Sum_{k=1..n} zk, s2 = Sum_{k=1..n} zk^2, s12 = (s1^2 - s2)/2, fk = t*(s12 - zk*(s1 - zk)) + zk*(s1 - zk) for k=1..n, P_n(t) = [(z1..zn)^2] Product_{k=1..n} fk. Then P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k), n >= 3. - Gheorghe Coserea, Dec 21 2018

A007105 Number of labeled Eulerian 3-regular digraphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 1, 44, 7570, 1975560, 749649145, 399035751464, 289021136349036, 277435664056527360, 345023964977303838105, 545099236551025860229460, 1075595203804151695555622446

Offset: 0

N. J. A. Sloane

nonn

Apparently this counts the binary variant (all elements in {0,1}) of A001500 with zero trace. - R. J. Mathar, May 16 2021

R. W. Robinson, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Óscar González, Carlos Beltrán, and Ignacio Santamaría, On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients, arXiv preprint arXiv:1301.6196 [cs.IT], 2013-2014. - From _N. J. A. Sloane_, Feb 19 2013
Jan Tóth and Ondřej Kuželka, Complexity of Weighted First-Order Model Counting in the Two-Variable Fragment with Counting Quantifiers: A Bound to Beat, arXiv:2404.12905 [cs.LO], 2024. See p. 18.

Cf. A001500, A284990.

A364068 Triangle T(n,k) read by rows: Number of traceless binary n X n matrices with all row and column sums equal to k, 1<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 9, 9, 1, 0, 44, 216, 44, 1, 0, 265, 7570, 7570, 265, 1, 0, 1854, 357435, 1975560, 357435, 1854, 1, 0, 14833, 22040361, 749649145, 749649145, 22040361, 14833, 1, 0, 133496, 1721632024

Offset: 1

R. J. Mathar, Jul 04 2023

			    0
    1        0
    2        1         0
    9        9         1      0
   44      216        44      1    0
  265     7570      7570    265    1 0
 1854   357435   1975560 357435 1854 1 0
14833 22040361 749649145

Cf. A000166 (k=1), A007107 (k=2), A284989 (see 1st col), A284990 (see 1st col, k=3), A007105 (k=3?), A284991 (see 1st col, k=4), A008300 (any trace)

T(n,n)=0. (k=n would require a 1 on the diagonal)
T(n,n-1)=1. (1 at all entries but the diagonal)
T(n,n-k) = T(n,k-1). (Flip entries 0<->1 and erase diagonal) - R. J. Mathar, Jul 26 2023

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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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A284989 Triangle T(n,k) read by rows: the number of n X n {0,1} matrices with trace k where each row sum and each column sum is 2.

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A007105 Number of labeled Eulerian 3-regular digraphs with n nodes.

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A364068 Triangle T(n,k) read by rows: Number of traceless binary n X n matrices with all row and column sums equal to k, 1<=k<=n.

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