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%I M1801 N0711 #106 Dec 22 2023 02:55:17
%S 1,2,7,36,317,5624,251610,33642660,14685630688,21467043671008,
%T 105735224248507784,1764356230257807614296,
%U 100455994644460412263071692,19674097197480928600253198363072,13363679231028322645152300040033513414,31735555932041230032311939400670284689732948
%N Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.
%C A diagonal of the array A(m,n) described in A028657. - _N. J. A. Sloane_, Sep 01 2013
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A002724/b002724.txt">Table of n, a(n) for n = 0..50</a> (terms 0..26 from Alois P. Heinz)
%H Manuel Kauers and Jakob Moosbauer, <a href="https://arxiv.org/abs/2006.01623">Good pivots for small sparse matrices</a>, arXiv:2006.01623 [cs.SC], 2020.
%H A. Kerber, <a href="/A002727/a002727.pdf">Experimentelle Mathematik</a>, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/22159/#881965">How many n-by-m binary matrices are there up to row and column permutations</a>
%H B. Misek, <a href="http://dml.cz/dmlcz/108444">On the number of classes of strongly equivalent incidence matrices</a>, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
%H Marko Riedel, <a href="/A002724/a002724.maple.txt">Maple code with two different algorithms</a>
%H M. Zivkovic, <a href="https://arxiv.org/abs/math/0511636">Classification of small (0,1) matrices</a>, arXiv:math/0511636 [math.CO], 2005.
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%H <a href="/index/Mat#inequiv">Index to number of inequivalent matrices modulo permutation of rows and columns</a>
%F 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
%F a(n) = A028657(2*n, n). - _Max Alekseyev_, Jun 24 2023
%p # See Marko Riedel link.
%t 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}]]]]];
%t 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]}];
%t A[n_, k_] := g[Min[n, k], Abs[n - k]];
%t Table[A[n, n], {n, 0, 15}] (* _Jean-François Alcover_, Aug 10 2018, after _Alois P. Heinz_ *)
%o (PARI) a(n) = A(n,n) \\ A defined in A028657. - _Andrew Howroyd_, Mar 01 2023
%Y Cf. A002623, A002727, A006148, A002728, A002725, A052269, A052271, A052272, A091059, A363845, A363846, A000595.
%Y Cf. A028657 (this sequence is the diagonal). - _N. J. A. Sloane_, Sep 01 2013
%Y Column k=2 of A246106.
%K nonn,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Feb 04 2000
%E a(15) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
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