A001501 -id:A001501 - cp's OEIS frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
;
1;
2 1;
5 2 1;
12 8 2 1;
33 22 8 2 1;
103 72 26 8 2 1;
333 229 87 26 8 2 1;
1183 782 295 92 26 8 2 1;
4442 2760 1036 315 92 26 8 2 1;
17576 10270 3735 1129 321 92 26 8 2 1;
72810 39770 13976 4117 1154 321 92 26 8 2 1;
314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..50
George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

More terms from Sean A. Irvine, Oct 02 2011

A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 90, 24, 1, 1, 120, 2040, 2040, 120, 1, 1, 720, 67950, 297200, 67950, 720, 1, 1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1, 1, 40320, 187530840, 24046189440, 116963796250, 24046189440, 187530840, 40320, 1, 1, 362880, 14398171200, 12025780892160, 315031400802720, 315031400802720, 12025780892160, 14398171200, 362880, 1

Offset: 0

N. J. A. Sloane

Or, triangle of multipermutation numbers T(n,k), n >= 0, 0 <= k <= n: number of relations on an n-set such that all vertical sections and all horizontal sections have k elements.

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       6,        1;
  1,   24,      90,       24,        1;
  1,  120,    2040,     2040,      120,       1;
  1,  720,   67950,   297200,    67950,     720,    1;
  1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,k).

Brendan D. McKay, Rows n = 0..30, flattened
C. J. Everett and P. R. Stein, The asymptotic number of integer stochastic matrices, Disc. Math. 1 (1971), 55-72.
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
Richard J. Mathar, Rencontres for equipartite distributions of multisets of colored balls into urns, vixra:2306.0157 (2023)
B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Brendan D. McKay, first 30 rows : entries named Bv[n,k,n,k]
Wouter Meeussen, relevant entries from B. D. McKay reference

Row sums give A067209.
Central coefficients are A058527.
Cf. A000142 (column 1), A001499 (column 2), A001501 (column 3), A058528 (column 4), A075754 (column 5), A172544 (column 6), A172541 (column 7), A172536 (column 8), A172540 (column 9), A172535 (column 11), A172534 (column 12), A172538 (column 13), A172537 (column 14).
Cf. A133687, A333157 (symmetric matrices), A257493 (nonnegative elements), A260340 (up to row permutations), A364068 (traceless).

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(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, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
} \\ Andrew Howroyd, Apr 03 2020

Comtet quotes Everett and Stein as showing that T(n,k) ~ (kn)!(k!)^(-2n) exp( -(k-1)^2/2 ) for fixed k as n -> oo.

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

More terms from Greg Kuperberg, Feb 08 2001

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

A000512 Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

Original entry on oeis.org

0, 0, 1, 1, 2, 7, 16, 51, 224, 1165, 7454, 56349, 481309, 4548786, 46829325, 519812910, 6177695783, 78190425826, 1049510787100, 14886252250208, 222442888670708, 3492326723315796, 57468395960854710, 989052970923320185, 17767732298980160822, 332572885090541084172, 6475438355244504235759, 130954580036269713385884

Offset: 1

Eric Rogoyski

Also, isomorphism classes of bicolored cubic bipartite graphs, where isomorphism cannot exchange the colors.

			n=4: every matrix with 3 1's in each row and column can be transformed by permutation of rows (or columns) into {1110,1101,1011,0111}, therefore a(4)=1. - _Michael Steyer_, Feb 20 2003

A. Burgess, P. Danziger, E. Mendelsohn, B. Stevens, Orthogonally Resolvable Cycle Decompositions, 2013; http://www.math.ryerson.ca/~andrea.burgess/OCD-submit.pdf
Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 284.

Index entries for sequences related to Latin squares and rectangles

Column k=3 of A133687.
Cf. A000186, A000513, A000840, A001501, A008325, A232215.
A079815 may be an erroneous version of this, or it may have a slightly different (as yet unknown) definition. - N. J. A. Sloane, Sep 04 2010.

Definition corrected by Brendan McKay, May 28 2006
a(1)-a(12) checked by Brendan McKay, Aug 27 2010
Terms a(15) and beyond from Andrew Howroyd, Apr 01 2020

A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 3, 14, 44, 129, 316, 714, 1452, 2775, 4963, 8478, 13838, 21827, 33306, 49504, 71754, 101871, 141807, 194128, 261570, 347633, 456026, 591384, 758596, 963657, 1212861, 1513806, 1874440, 2304225, 2813030, 3412466, 4114608, 4933519

Offset: 0

Vladeta Jovovic, Nov 24 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Illustration of initial terms
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 3 of A318951.
Cf. A002817, A052282, A058390, A058391, A058392.
Cf. A001501, A050535, A050913, A058528, A058783, A058784, A058785.

a[n_] := (m = Mod[n, 6]; (n^3 + 9*n^2 + 39*n + 120)*n^3 + Which[m == 0, 12*(23*n^2 + 32*n + 24), m == 1 || m == 5, 249*n^2 + 303*n + 143, m == 2 || m == 4, 4*(69*n^2 + 96*n + 56), m == 3, 3*(83*n^2 + 101*n + 69)])/288; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 12 2011, after Vladeta Jovovic *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(3, 3, n); \\ Andrew Howroyd, Sep 05 2018

a(n) = (1/6)*(C(C(n + 2, 2) + 2, 3) + 3/2*floor((n + 2)/2)*(C(n + 2, 2) - floor((n + 2)/2)) + 3*C(floor((n + 2)/2) + 2, 3) + 2*floor(C(n + 2, 2)/3) + 2*C(C(n + 2, 2) - 3*floor(C(n + 2, 2)/3) + 2, 3)).

Empirical G.f.: -(x^8 + 3*x^7 + 14*x^6 + 12*x^5 + 15*x^4 + 9*x^3 + 5*x^2 + 1) / ((x-1)^7*(x+1)^3*(x^2+x+1)). - Colin Barker, Dec 27 2012

More terms from Marc LeBrun, Dec 11 2000

A185178 Number of distinct values of the permanent of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 32, 61, 109

Offset: 3

Vladimir Shevelev, Feb 19 2011

The values a(n) are unknown for n>=9.

a(12) >= 170, a(13) >= 276, a(14) >= 438, a(15) >= 547. - Robert P. P. McKone, Jul 14 2025

V. I. Bolshakov, On spectrum of permanent on Lambda_n^k, Proc. of Seminar on Discrete Math. and Appl., Moscow State Univ. (1986), 65-73 (in Russian).

Robert P. P. McKone, One example matrix for each known permanent for a(3)..a(15), with a(11)..a(15) being incomplete
Robert P. P. McKone, Complete frequence counts for each permanent for a(3)..a(11)
Robert P. P. McKone, Robert P. P. McKone, C++ code for enumerating and counting each permanent's frequency for a(n)
V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317 (see pp. 2307-2309).

Cf. A185179 (values of the permanent), A176211, A176212.

Cf. A001501 (number of n X n (0,1)-matrix with exactly three 1's in each row and each column).

a(9)-a(11) from Robert P. P. McKone, Jul 22 2025

A284990 Triangle T(n,t) read by rows: the number of n X n {0,1} matrices with trace t where each row sum and each column sum is 3.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 8, 9, 44, 210, 420, 610, 540, 216, 7570, 33120, 66870, 82080, 66870, 33120, 7570, 1975560, 8171730, 15729000, 18433415, 14372820, 7499940, 2398900, 357435, 749649145, 2971510080, 5508175260, 6267658544, 4815171270, 2570369760, 932429820, 209185200, 22040361

Offset: 0

R. J. Mathar, Apr 07 2017

			0:        1
1:        0       0
2:        0       0        0
3:        0       0        0        1
4:        1       0        6        8        9
5:       44     210      420      610      540     216
6:     7570   33120    66870    82080    66870   33120    7570
7:  1975560 8171730 15729000 18433415 14372820 7499940 2398900 357435

Alois P. Heinz, Rows n = 0..11, flattened
Index entries for sequences related to binary matrices

Cf. A007107 (diagonal?), A001501 (row sums), A007105 (column 0?), A110040 (symmetric matrices).

More terms from Alois P. Heinz, Apr 09 2017

A058527 Number of 2n X 2n 0-1 matrices with n ones in each row and each column.

Original entry on oeis.org

1, 2, 90, 297200, 116963796250, 6736218287430460752, 64051375889927380035549804336, 108738182111446498614705217754614976371200, 34812290428176298285394893936773707951192224124239796250, 2188263032066768922535710968724036448759525154977348944382853301460850000

Offset: 0

David desJardins, Dec 22 2000

nonn

Vladeta Jovovic, Nov 12 2006, Table of n, a(n) for n = 0..15
A Conflitti, C. M. Da Fonseca, and R. Mamede, The maximal length of a chain in the Bruhat order for a chain of binary matrices., Lin. Algebra Applic. (2011)
Alessandro Conflitti, C. M. da Fonseca and Ricardo Mamede, On the largest size of an antichain in the Bruhat order for A(2k, k).
Alessandro Conflitti, C. M. da Fonseca and Ricardo Mamede, On the Largest Size of an Antichain in the Bruhat Order for A(2k,k), ORDER, 2011, DOI: 10.1007/s11083-011-9241-1.
Jonathan Jedwab and Tabriz Popatia, A new representation of mutually orthogonal frequency squares, Simon Fraser University (Burnaby, BC, Canada, 2020).
M. A. Khojastepour and M. Farajzadeh-Tehrani, Characterizing per Node Degrees of Freedom in an Interference Network, 2014; also 2014 IEEE International Symposium on Information Theory, pp. 1016-1020.
B. D. McKay, 0-1 matrices with constant row and column sums
Michael Penn, A not so magic square..., YouTube video, 2021.
Wikipedia, Dynamic programming

Central coefficients of A008300.
Main diagonal of A376935.
Cf. A001499, A001501, A253316.

More terms (using dynamic programming in Python) from Greg Kuperberg, Feb 08 2001

More terms from Vladeta Jovovic, Nov 12 2006

A181344 Number of n X n matrices over {0,1} with rows and columns summing to 3, rows and columns sorted (>=) by value.

Original entry on oeis.org

0, 0, 1, 1, 5, 25, 161, 1112, 8787, 76156, 728699, 7609065, 86162795, 1050755884, 13728407061, 191309852944

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Oct 14 2010

			n=4: {1110,1101,1011,0111} is the only matrix where each row (column) - read as a binary number - is equal to or larger than the previous one, so a(4)=1.

Bert Dobbelaere, Illustration of initial terms

Cf. A000512, A001501, A181345.

a(10)-a(16) from Bert Dobbelaere, Feb 23 2020

A181345 Number of n X n matrices over {0,1} with rows and columns summing to 3, rows and columns sorted (>) by value.

Original entry on oeis.org

0, 0, 0, 1, 2, 12, 87, 662, 5611, 51141, 509277, 5504398, 64122940, 800741192, 10673478573, 151323048909

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Oct 18 2010

			n=4: {1110,1101,1011,0111} is the only matrix where each row (column) - read as a binary number - is larger than the previous one, so a(4)=1.

Bert Dobbelaere, Illustration of initial terms

Cf. A000512, A001501, A181344.

Offset corrected and a(10)-a(16) from Bert Dobbelaere, Feb 23 2020

cp's OEIS Frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

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A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

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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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A000512 Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

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A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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Mathematica

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A185178 Number of distinct values of the permanent of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

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A284990 Triangle T(n,t) read by rows: the number of n X n {0,1} matrices with trace t where each row sum and each column sum is 3.

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A058527 Number of 2n X 2n 0-1 matrices with n ones in each row and each column.