A002727 -id:A002727 - cp's OEIS frontend

A002623 Expansion of 1/((1-x)^4*(1+x)).

1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

N. J. A. Sloane

Also a(n) is the number of nondegenerate triangles that can be made from rods of lengths 1 to n+1. - Alfred Bruckstein; corrected by Hans Rudolf Widmer, Nov 02 2023
Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,...,n. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
Also number of 2 X n binary matrices up to row and column permutation (see the link: Binary matrices up to row and column permutations). - Vladeta Jovovic
Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 (cf. A002717, also the Larsen article). - Henry Bottomley, Aug 08 2000
Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy, Oct 13 2003
Also Molien series for certain 4-D representation of cyclic group of order 2. - N. J. A. Sloane, Jun 12 2004
From Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: (Start)
a(n) = floor( (n+2)*(n+4)*(2n+3) / 24 ). E.g., a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:
     /\
    /\/\
   /\/\/\
  /\/\/\/\
(End)
Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - Michael Somos, May 12 2005
a(n) = A108561(n+4,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
Also number of nonisomorphic planes with n points and 2 lines. E.g., a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e., up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
a(n-2) is the number of ways to pick two non-overlapping subwords of equal nonzero length from a word of length n. E.g., a(5-2)=a(3)=13 since the word 12345 of length 5 has the following subword pairs: 1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3,5; 4,5; 12,34; 12,45; 23,45. - Michael Somos, Oct 22 2006
Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
From Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007: (Start)
Also number of squares of any size in a staircase of n steps built with unit squares:
   
  ||__
  ||__|
  ||__||
For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, hence c(3)=7.
Columns sums of:
  1 3 6 10 15 21 28 ...
      1  3  6 10 15 ...
            1  3  6 ...
                  1 ...
  ---------------------
  1 3 7 13 22 34 50 ...
(End)
a(n) = sum of row n+1 of triangle A134446. Also, binomial transform of [1, 2, 2, 0, 1, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007
Let b(n) be the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and 2w=x+y+z+n; then b(n+3) = a(n) for n >= 0. - Clark Kimberling, May 08 2012
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w >= x+y and x <= y. - Clark Kimberling, Jun 04 2012
Also, number of unlabeled bipartite graphs with two left vertices and n right vertices. - Yavuz Oruc, Jan 14 2018
Also number of triples (x,y,z) with 0 < x <= y <= z <= n + 1, x + y > z. - Ralf Steiner, Feb 06 2020
Bisections A002412 and A016061: a(2*k) = k*(k+1)*(4*k-1)/3! and a(2*k+1) = (k+1)*(k+2)*(4*k+9)/3!, for k >= 0. See the Woolhouse link, II. Solution by Stephen Watson, p. 65, with index shifts. - Mo Li, Apr 02 2020
Also, Wiener index of the square of the path graph P_(n+2). - Allan Bickle, Aug 01 2020
Maximum Wiener index of all maximal 2-degenerate graphs with n+2 vertices.  (A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.)  The extremal graphs are squares of paths, so the bound also applies to 2-trees and maximal outerplanar graphs. - Allan Bickle, Sep 15 2022

			G.f. = 1 + 3*x + 7*x^2 + 13*x^3 + 22*x^4 + 34*x^5 + 50*x^6 + 70*x^7 + 95*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdős is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Atmaca and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 5th line.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!), arXiv preprint arXiv:1511.06791 [math.CO], 2015.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in Arithmetic Progressions, arXiv preprint arXiv:1301.5122 [math.NT], 2013.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 203
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 413
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]. See page 79.
W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
Math StackExchange, cycle index for S_2 X S_4, Apr. 2021
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218. See page 217.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Giovanni Resta, Illustration for a(8)=70.
Marko Riedel, The O.g.f. of inequivalent colorings of a 2xN board with C colors in N is (1/2) 1/(1-w)^(C^2) + (1/2) 1/(1+w)^(C(C-1)/2)*1/(1-w)^(C(C+1)/2) by PET
Sebastian Sassi, Aula Al-Adulrazzaq, Matti Heikinheimo, and Kimmo Tuominen, Fast numerical evaluation of dark matter direct detection event rates, arXiv:2504.19714 [hep-ph], 2025. See p. 7.
Atsuto Seko, Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems, J. Appl. Phys. (2023) Vol. 133, 011101.
J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.
Eric Weisstein's World of Mathematics, Triangle Counting.
W. S. B. Woolhouse, Problem 2420. On the probability of the number of triangles, Mathematical questions with their solutions, v. 9 (June 1868), pp. 63-65.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002620 (first differences), A000292, A001752 (partial sums), A062109 (binomial transf.).
Bisections A002412, A016061.
Cf. also A002717 (a companion sequence), A002727, A006148, A057524, A134446, A014125, A122046, A122047.
Cf. A000217, A006918, A058187, A108561.
The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623 (this sequence), A014125, A122046, A122047, A175724, respectively.

A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,3)/2;
seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); # Lewis
a := n -> ((-1)^n*3 + 45 + 68*n + 30*n^2 + 4*n^3) / 48:
seq(a(n), n=0..46); # Peter Luschny, Jan 22 2018

CoefficientList[Series[1/((1-x)^3(1-x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,3,7,13,22},50] (* Harvey P. Dale, Jul 19 2011 *)
Table[((2 n^3 + 15 n^2 + 34 n + 45 / 2 + (3/2) (-1)^n) / 24), {n, 0, 100}] (* Vincenzo Librandi, Jan 15 2018 *)
a[ n_] := Floor[(n + 2)*(n + 4)*(2*n + 3)/24]; (* Michael Somos, Feb 19 2024 *)

{a(n) = (8 + 34/3*n + 5*n^2 + 2/3*n^3) \ 8}; /* Michael Somos, Sep 04 1999 */

x='x+O('x^50); Vec(1/((1 - x)^3 * (1 - x^2))) \\ Indranil Ghosh, Apr 04 2017

def A002623(n): return ((n+2)*(n+4)*((n<<1)+3)>>3)//3 # Chai Wah Wu, Mar 25 2024

a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
a(n)+a(n+1) = A000292(n+1).
a(n) = a(n-2) + A000217(n+1) = A002717(n+2) - A000292(n+1).
Also: a(n) = C(n+3, 3) - a(n-1) with a(0)=1. - Labos Elemer, Apr 26 2003
From Paul Barry, Jul 01 2003: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+3,3).
The signed version 1, -3, 7, ... has the formula:
a(n) = (4*n^3 + 30*n^2 + 68*n + 45)*(-1)^n/48 + 1/16.
This is the partial sums of the signed version of A000292. (End)
From Paul Barry, Jul 21 2003: (Start)
a(n) = Sum_{k=0..n} floor((k+2)^2/4).
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (1+(-1)^i)/2. (End)
a(n) = a(n - 2) + (n*(n - 1))/2, with n>2, a(1)=0, a(2)=1; a(n) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48, with offset 2. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004 (formula simplified by Bruno Berselli, Aug 29 2013)
a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, with offset 1. - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
A002620(n+3) = a(n+1) - a(n). - Michael Somos, Sep 04 1999
Euler transform of length 2 sequence [ 3, 1]. - Michael Somos, Sep 04 2006
a(n) = -a(-5-n) for all n in Z. - Michael Somos, Sep 04 2006
Let P(i,k) be the number of integer partitions of n into k parts, then with k=2 we have a(n) = sum_{m=1}^{n} sum_{i=k}^{m} P(i,k). For k=1 we get A000217 = triangular numbers. - Thomas Wieder, Feb 18 2007
a(n) = (n+(3+(-1)^n)/2)*(n+(7+(-1)^n)/2)*(2*n+5-2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007 (corrected by Bruno Berselli, Aug 30 2013)
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A006918(n+1) + A006918(n).
a(n) = A058187(n-2) + 2*A058187(n-1) + A058187(n). (End)
a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22; for n > 4, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Harvey P. Dale, Jul 19 2011
a(n) = Sum_{i=0..n+2} floor(i/2)*ceiling(i/2). - Bruno Berselli, Aug 30 2013
a(n) = 15/16 + (1/16)*(-1)^n + (17/12)*n + (5/8)*n^2 + (1/12)*n^3. - Robert Israel, Jul 07 2014
a(n) = Sum_{i=0..n+2} (n+1-i)*floor(i/2+1). - Bruno Berselli, Apr 04 2017
a(n) = 1 + floor((2*n^3 + 15*n^2 + 34*n) / 24). - Allan Bickle, Aug 01 2020
E.g.f.: ((24 + 51*x + 21*x^2 + 2*x^3)*cosh(x) + (21 + 51*x + 21*x^2 + 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021

A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

Original entry on oeis.org

1, 2, 7, 36, 317, 5624, 251610, 33642660, 14685630688, 21467043671008, 105735224248507784, 1764356230257807614296, 100455994644460412263071692, 19674097197480928600253198363072, 13363679231028322645152300040033513414, 31735555932041230032311939400670284689732948

Offset: 0

N. J. A. Sloane

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Also, number of bipartite graphs with both partite sets of size n, one of which is marked. For connected bipartite graphs, see A363846. - Max Alekseyev, Jun 24 2023

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from Alois P. Heinz)
Manuel Kauers and Jakob Moosbauer, Good pivots for small sparse matrices, arXiv:2006.01623 [cs.SC], 2020.
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
Mathematics Stack Exchange, How many n-by-m binary matrices are there up to row and column permutations
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Marko Riedel, Maple code with two different algorithms
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Index entries for sequences related to binary matrices
Index to number of inequivalent matrices modulo permutation of rows and columns

Cf. A002623, A002727, A006148, A002728, A002725, A052269, A052271, A052272, A091059, A363845, A363846, A000595.
Cf. A028657 (this sequence is the diagonal). - N. J. A. Sloane, Sep 01 2013
Column k=2 of A246106.

Maple
```
# See Marko Riedel link.
```

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Union[Flatten[Table[ Function[{p}, p + j*x^i] /@ b[n - i*j, i - 1], {j, 0, n/i}]]]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
Table[A[n, n], {n, 0, 15}] (* Jean-François Alcover, Aug 10 2018, after Alois P. Heinz *)

a(n) = A(n,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fixA[...] = 2^Sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Christian G. Bower, Dec 18 2003

a(n) = A028657(2*n, n). - Max Alekseyev, Jun 24 2023

More terms from Vladeta Jovovic, Feb 04 2000

a(15) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

A028657 Triangle read by rows: T(n,k) = number of n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 22, 36, 22, 6, 1, 1, 7, 34, 87, 87, 34, 7, 1, 1, 8, 50, 190, 317, 190, 50, 8, 1, 1, 9, 70, 386, 1053, 1053, 386, 70, 9, 1, 1, 10, 95, 734, 3250, 5624, 3250, 734, 95, 10, 1, 1, 11, 125, 1324, 9343, 28576, 28576, 9343, 1324, 125, 11, 1

Offset: 0

Vladeta Jovovic, Jun 16 2000

Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color; the color classes are not interchangeable.
Also the number of principal transversal matroids (also known as fundamental transversal matroids) of size n and rank k (originally enumerated by Brylawski). - Gordon F. Royle, Oct 30 2007
This sequence is also obtained if we read the array A(m,n) = number of inequivalent m X n binary matrices by antidiagonals, where equivalence means permutations of rows or columns (m>=0, n>=0) [Kerber]. - N. J. A. Sloane, Sep 01 2013

			The triangle T(n,k) begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 22, 36, 22,  6,  1;
  ...
For example, there are 36 graphs on 6 nodes with a distinguished bipartite block with 3 nodes.
The array A(m,n) (m>=0, n>=0) (see Comments) begins:
  1 1  1    1     1      1        1         1           1 ...
  1 2  3    4     5      6        7         8           9 ...
  1 3  7   13    22     34       50        70          95 ...
  1 4 13   36    87    190      386       734        1324 ...
  1 5 22   87   317   1053     3250      9343       25207 ...
  1 6 34  190  1053   5624    28576    136758      613894 ...
  1 7 50  386  3250  28576   251610   2141733    17256831 ...
  1 8 70  734  9343 136758  2141733  33642660   508147108 ...
  1 9 95 1324 25207 613894 17256831 508147108 14685630688 ...
... - _N. J. A. Sloane_, Sep 01 2013

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Alois P. Heinz, Rows n = 0..45, flattened (first 20 rows from R. W. Robinson)
Thomas H. Brylawski, An affine representation for transversal geometries, Studies in Appl. Math. 54 (1975), no. 2, 143-160.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 1.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 1.
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Carlos Simpson, Learning proofs for the classification of nilpotent semigroups, arXiv:2106.03015 [cs.LG], 2021.
Index to OEIS entries related to number of inequivalent matrices modulo permutation of row and columns.

Row sums give A049312.
A246106 is a very similar array.
Cf. A055080, A049312, A052265, A242093.
Diagonals of the array A(m,n) give A002724, A002725, A002728.
Rows (or columns) give A002623, A002727, A006148, A052264.
A(n,k) = A353585(2, n, k).

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
      coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
    end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Union[ Flatten[ Table[ Function[ {p}, p + j*x^i] /@ b[n - i*j, i-1], {j, 0, n/i}]]]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[ Sum[GCD[i, j] * Coefficient[s, x, i] * Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[i^Coefficient[s, x, i] * Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i] * Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
A(n, m)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(exp(sum(t=1, n, 2^K(q, t)/t*x^t) + O(x*x^n)), n)); s/m!}
{ for(r=0, 10, for(k=0, r, print1(A(r-k,k), ", ")); print) } \\ Andrew Howroyd, Mar 25 2020

\\ G(k,x) gives k-th column as rational function (see Jovovic link).
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
Fix(q,x)={my(v=divisors(lcm(Vec(q))), u=apply(t->2^sum(j=1, #q, gcd(t, q[j])), v)); 1/prod(i=1, #v, my(t=v[i]); (1-x^t)^(sum(j=1, i, my(d=t/v[j]); if(!frac(d), moebius(d)*u[j]))/t))}
G(m,x)={my(s=0); forpart(q=m, s+=permcount(q)*Fix(q,x)); s/m!}
T(n,k)={my(m=max(k, n-k)); polcoef(G(n-m, x + O(x*x^m)), m)} \\ Andrew Howroyd, Mar 26 2020

A028657(n,k)=A353585(2, n, k) \\ M. F. Hasler, May 01 2022

A(m,n) = Sum_{p in P(m), q in P(n)} 2^Sum_{i in p, j in q} gcd(i,j) / (N(p) N(q)) where P(m) are the partition of m (see e.g., A036036), N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. [corrected by Anders Kaseorg, Oct 04 2024]

A006148 Number of 4 X n binary matrices up to row and column permutations.

Original entry on oeis.org

1, 5, 22, 87, 317, 1053, 3250, 9343, 25207, 64167, 155004, 357009, 787586, 1670643, 3419552, 6774765, 13027340, 24372942, 44462456, 79240762, 138204782, 236258358, 396409924, 653639898, 1060379169, 1694174350, 2668300758, 4146300078, 6361709115, 9644583474

Offset: 0

N. J. A. Sloane

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Index entries for linear recurrences with constant coefficients, signature (6, -12, 6, 6, -6, 22, -54, 33, -4, 12, 60, -125, 54, -54, 70, 87, -132, 64, -132, 87, 70, -54, 54, -125, 60, 12, -4, 33, -54, 22, -6, 6, 6, -12, 6, -1).
Index entries for two-way infinite sequences

Cf. A002623, A002727, A006380.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

CoefficientList[Series[(x^20 - x^19 + 4 x^18 + 9 x^17 + 23 x^16 + 39 x^15 + 90 x^14 + 131 x^13 + 204 x^12 + 238 x^11 + 252 x^10 + 238 x^9 + 204 x^8 + 131 x^7 + 90 x^6 + 39 x^5 + 23 x^4 + 9 x^3 + 4 x^2 - x + 1)/((1 - x^4)^3 (1 - x^3)^4 (1 - x^2)^3 (1 - x)^6), {x, 0, 45}], x] (* Vincenzo Librandi, Oct 13 2015 *)
LinearRecurrence[{6,-12,6,6,-6,22,-54,33,-4,12,60,-125,54,-54,70,87,-132,64,-132,87,70,-54,54,-125,60,12,-4,33,-54,22,-6,6,6,-12,6,-1},{1,5,22,87,317,1053,3250,9343,25207,64167,155004,357009,787586,1670643,3419552,6774765,13027340,24372942,44462456,79240762,138204782,236258358,396409924,653639898,1060379169,1694174350,2668300758,4146300078,6361709115,9644583474,14456861538,21439125178,31471971903,45755970759,65915132560,94129925265},30] (* Harvey P. Dale, Jun 22 2021 *)

Vec(G(4, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^20 - x^19 + 4*x^18 + 9*x^17 + 23*x^16 + 39*x^15 + 90*x^14 + 131*x^13 + 204*x^12 + 238*x^11 + 252*x^10 + 238*x^9 + 204*x^8 + 131*x^7 + 90*x^6 + 39*x^5 + 23*x^4 + 9*x^3 + 4*x^2 - x + 1)/((1 - x^4)^3*(1 - x^3)^4*(1 - x^2)^3*(1 - x)^6). - Vladeta Jovovic, Feb 04 2000

More terms from Vladeta Jovovic, Feb 04 2000
Definition corrected by Max Alekseyev, Feb 05 2010
More terms from Vincenzo Librandi, Oct 13 2015

A005783 Number of 3-covers of an unlabeled n-set.

Original entry on oeis.org

1, 3, 9, 23, 51, 103, 196, 348, 590, 960, 1506, 2290, 3393, 4905, 6945, 9651, 13185, 17739, 23542, 30846, 39954, 51206, 64986, 81730, 101935, 126141, 154967, 189093, 229269, 276325, 331182, 394830, 468372, 553002, 650016, 760824, 886963

Offset: 0

N. J. A. Sloane

Equals first differences of A002727. - Vladeta Jovovic, May 24 2000

Number of 3 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

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

Stefano Spezia, Table of n, a(n) for n = 0..10000 (terms for n = 1..1000 from T. D. Noe)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Masaaki Harada, Ken Saito, Binary linear complementary dual codes, arXiv:1802.06985 [math.CO], 2018.
Vladeta Jovovic, Binary matrices up to row and column permutations
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,-1,3,6,-6,-3,1,3,1,-3,1).

Column 3 of A055080.

Cf. A002727, A002620, A005784, A005785, A005786, A055066.

CoefficientList[Series[(x^6+x^4+2x^3+x^2+1)/((1-x^3)^2(1-x^2)^2 (1-x)^3),{x,0,50}],x] (* Harvey P. Dale, May 19 2011 *)

Vec(G(3, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^6+x^4+2*x^3+x^2+1)/((1-x^3)^2*(1-x^2)^2*(1-x)^3).
a(n) ~ n^6/4320. - Stefano Spezia, Aug 08 2022
a(n) = n^6/4320 + 7*n^5/1440 + 79*n^4/1728 + 35*n^3/144 + 2939*n^2/4320 + 8863*n/8640 + 1 + (n/16 + 7/32)*floor(n/2) + (n/9 + 11/27)*floor(n/3) + floor((n+1)/3)/27. - Vaclav Kotesovec, Aug 09 2022

More terms from Vladeta Jovovic, May 24 2000

a(0) = 1 prepended by Stefano Spezia, Aug 09 2022

A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 4, 7, 19, 32, 68, 114, 210, 336, 562, 862, 1349, 1987, 2950, 4201, 5991, 8278, 11422, 15386, 20660, 27218, 35718, 46158, 59401, 75475, 95494, 119545, 149035, 184118, 226562, 276620, 336470, 406490, 489344, 585572, 698397, 828549, 979896

Offset: 0

N. J. A. Sloane

Also the number of ways in which to label the vertices of the cube (or faces of the octahedron) with nonnegative integers summing to n, where labelings that differ only by rotation or reflection are considered the same. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004

			Representatives of the seven classes of 3 X 3 binary matrices are:
[ 1 1 1 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 1 ] [ 0 1 1 ] [ 0 1 1 ] [ 0 1 1 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 0 ] [ 1 0 0 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 1 1 ] [ 1 0 0 ].

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1052.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,2,-5,7,-4,2,2,-8,10,-8,2,2,-4,7,-5,2,-1,-2,3,-1).
Index entries for sequences related to binary matrices
Index entries for two-way infinite sequences

Column k=3 of A363349.

Cf. A000601, A005232, A006382, A006380, A002727, A006148.

Vec((1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48 + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48.

G.f.: (x^14 - 2*x^13 + 3*x^12 - 2*x^11 + 5*x^10 - 4*x^9 + 7*x^8 - 4*x^7 + 7*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 2*x + 1)/(x^6 - 1)/(x^2 + 1)^2/(x^2 + x + 1)/(x + 1)^3/(x - 1)^7.

Entry revised by Vladeta Jovovic, Aug 05 2000

Definition corrected by Max Alekseyev, Feb 05 2010

A006382 Number of n X 4 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 5, 11, 41, 101, 301, 757, 1981, 4714, 11133, 24763, 53818, 111941, 226857, 444260, 848620, 1576226, 2862426, 5077454, 8827758, 15043096, 25183794, 41434222, 67108437, 107051463, 168402958, 261384026, 400684767, 606936536

Offset: 0

N. J. A. Sloane

nonn

			Representatives of the five classes of 2 X 4 binary matrices are:
[ 1 1 1 1 ] [ 1 1 1 0 ] [ 1 1 0 1 ] [ 1 0 1 1 ] [ 0 1 1 1 ]
[ 1 1 1 1 ] [ 1 1 1 1 ] [ 1 1 1 0 ] [ 1 1 0 0 ] [ 1 0 0 0 ]

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -13, 10, 4, -14, 25, -46, 53, -36, 8, 44, -111, 138, -123, 106, -54, -66, 181, -238, 259, -220, 98, 36, -150, 280, -352, 280, -150, 36, 98, -220, 259, -238, 181, -66, -54, 106, -123, 138, -111, 44, 8, -36, 53, -46, 25, -14, 4, 10, -13, 6, -1).
Index entries for sequences related to binary matrices
Index entries for two-way infinite sequences

Column k=4 of A363349.

Cf. A005232, A006380, A006381, A002727, A006148.

G.f. : (1/(1 - x^1)^16 + 51/(1 - x^2)^8 + 12/(1 - x^1)^8/(1 - x^2)^4 + 84/(1 - x^4)^4 + 12/(1 - x^1 )^4/(1 - x^2)^6 + 32/(1 - x^1)^4/(1 - x^3)^4 + 96/(1 - x^2)^2/(1 - x^6)^2 + 48/(1 - x^1)^2/(1 - x^2)^1/(1 - x^4)^3 + 48/(1 - x^8)^2)/384.

Entry revised by Vladeta Jovovic, Aug 05 2000

A055609 Number of 3 X n binary matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 5, 17, 42, 91, 180, 328, 565, 930, 1470, 2248, 3344, 4849, 6881, 9579, 13104, 17649, 23442, 30736, 39833, 51074, 64842, 81574, 101766, 125959, 154771, 188883, 229044, 276085, 330926, 394558, 468083, 552696, 649692, 760482, 886602, 1029691, 1191539, 1374065, 1579326

Offset: 1

Vladeta Jovovic, Jun 03 2000

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -3, -1, 3, 6, -6, -3, 1, 3, 1, -3, 1).

Column k=3 of A056152.

Cf. A024206, A002623, A002727.

Vec((G(3, x) - G(2, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: x*(x^8-x^7-x^6-2*x^5+2*x^4+x^3-3*x^2-2*x-1)/((x^3-1)^2*(x^2-1)^2*(x-1)^3).

Terms a(37) and beyond from Andrew Howroyd, Mar 25 2020

A052264 Number of 5 X n binary matrices up to row and column permutations.

Original entry on oeis.org

1, 6, 34, 190, 1053, 5624, 28576, 136758, 613894, 2583164, 10208743, 38013716, 133872584, 447620002, 1426354541, 4346885204, 12710830673, 35768703586, 97125981825, 255111287298, 649598148384, 1606754306778, 3867515638005, 9074220508038, 20784247213232

Offset: 0

Vladeta Jovovic, Feb 04 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Binary matrices up to row and column permutations
Míšek [Misek], Bohuslav: O počtu tříd silně ekvivalentních incidenčních matic. (Czech) [On the number of classes of strongly equivalent incidence matrices]. Časopis pro pěstování matematiky, vol. 89 (1964), issue 2, pp. 211-218.
Index entries for linear recurrences with constant coefficients, signature (8, -24, 30, -6, -18, 27, -60, 87, -108, 147, -36, -82, 8, -147, 260, -253, 672, -413, -14, -471, -270, 612, -330, 2024, -1042, 213, -2022, -423, -18, 600, 4032, -858, 1468, -4952, -714, -3255, 1722, 5577, 1638, 4032, -5862, -1352, -8594, 1530, 3114, 5619, 6306, -2324, -170, -11814, -170, -2324, 6306, 5619, 3114, 1530, -8594, -1352, -5862, 4032, 1638, 5577, 1722, -3255, -714, -4952, 1468, -858, 4032, 600, -18, -423, -2022, 213, -1042, 2024, -330, 612, -270, -471, -14, -413, 672, -253, 260, -147, 8, -82, -36, 147, -108, 87, -60, 27, -18, -6, 30, -24, 8, -1).

Cf. A002623, A002727, A006148.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Vec(G(5, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^68 - 2*x^67 + 10*x^66 + 32*x^65 + 175*x^64 + 794*x^63 + 3441*x^62 + 13186*x^61 + 46027*x^60 + 146118*x^59 + 427347*x^58 + 1155432*x^57 + 2912873*x^56 + 6875608*x^55 + 15281029*x^54 + 32094658*x^53 + 63945531*x^52 + 121210914*x^51 + 219194198*x^50 + 378998758*x^49 + 627863648*x^48 + 998282344*x^47 + 1525746624*x^46 + 2244502676*x^45 + 3181886869*x^44 + 4351201210*x^43 + 5744918381*x^42 + 7328807372*x^41 + 9039504349*x^40 + 10785767638*x^39 + 12455264802*x^38 + 13925287384*x^37 + 15077477135*x^36 + 15812782150*x^35 + 16065602576*x^34 + 15812782150*x^33 + 15077477135*x^32 + 13925287384*x^31 + 12455264802*x^30 + 10785767638*x^29 + 9039504349*x^28 + 7328807372*x^27 + 5744918381*x^26 + 4351201210*x^25 + 3181886869*x^24 + 2244502676*x^23 + 1525746624*x^22 + 998282344*x^21 + 627863648*x^20 + 378998758*x^19 + 219194198*x^18 + 121210914*x^17 + 63945531*x^16 + 32094658*x^15 + 15281029*x^14 + 6875608*x^13 + 2912873*x^12 + 1155432*x^11 + 427347*x^10 + 146118*x^9 + 46027*x^8 + 13186*x^7 + 3441*x^6 + 794*x^5 + 175*x^4 + 32*x^3 + 10*x^2 - 2*x + 1)/((x^6 - 1)^2*(x^4 + x^3 + x^2 + x + 1)^6*(x^3 - x^2 + x - 1)^6 * (x^2 + x + 1)^6*(x + 1)^10*(x - 1)^24).

Name clarified by Ching Pong Siu, Aug 30 2022

A002725 Number of incidence matrices: n X (n+1) binary matrices under row and column permutations.

Original entry on oeis.org

1, 3, 13, 87, 1053, 28576, 2141733, 508147108, 402135275365, 1073376057490373, 9700385489355970183, 298434346895322960005291, 31479360095907908092817694945, 11474377948948020660089085281068730, 14568098446466140788730090352230460100956

Offset: 0

N. J. A. Sloane

a(0) = 1 by convention.

			a(1) = 3: [0,0], [0,1], [1,1].
a(2) = 13:
000 000 000 000 001 001 001 001 001 011 011 011 111
000 001 011 111 001 010 011 110 111 011 101 111 111

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..23 from Alois P. Heinz)
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.

Cf. A002623, A002727, A006148, A002728, A002724.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*
      coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/
      mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/
      mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),
      t=b(n+1$2)), s=b(n$2)):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
a[n_] := Sum[Sum[2^Sum[ Sum [ GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[ i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+1, n+1]}], {s,  b[n, n]}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

a(n) = A(n+1,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+1} (fix A[s_1, s_2, ...; t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fix A[...] = 2^sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Sean A. Irvine, Jul 31 2014

More terms from Vladeta Jovovic, Feb 04 2000

cp's OEIS Frontend

A002623 Expansion of 1/((1-x)^4*(1+x)).

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A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

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A028657 Triangle read by rows: T(n,k) = number of n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

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A006148 Number of 4 X n binary matrices up to row and column permutations.

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A005783 Number of 3-covers of an unlabeled n-set.

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A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

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