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%I #107 Dec 29 2022 06:03:09
%S 1,2,16,512,65536,33554432,68719476736,562949953421312,
%T 18446744073709551616,2417851639229258349412352,
%U 1267650600228229401496703205376,2658455991569831745807614120560689152,22300745198530623141535718272648361505980416,748288838313422294120286634350736906063837462003712
%N a(n) = 2^(n^2).
%C For n >= 1, a(n) is the number of n X n (0, 1) matrices.
%C Also number of directed graphs on n labeled nodes allowing self-loops (cf. A053763).
%C 1/2^(n^2) is the Hankel transform of C(n, n/2)*(1 + (-1)^n)/(2*2^n), or C(2n, n)/4^n with interpolated zeros. - _Paul Barry_, Sep 27 2007
%C Hankel transform of A064062. - _Philippe Deléham_, Nov 19 2007
%C a(n) is also the order of the semigroup (monoid) of all binary relations on an n-set. - _Abdullahi Umar_, Sep 14 2008
%C With offset = 1, a(n) is the number of n X n (0, 1) matrices with an even number of 1's in every row and in every column. - _Geoffrey Critzer_, May 23 2013
%C a(n) is the number of functions from an n-set to its power set (by definition of function including the empty function only when n = 0). - _Rick L. Shepherd_, Dec 27 2014
%D John M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). - _Abdullahi Umar_, Sep 14 2008
%H Vincenzo Librandi, <a href="/A002416/b002416.txt">Table of n, a(n) for n = 0..33</a>
%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Theresia Eisenkölbl, <a href="https://arxiv.org/abs/math/0106038">2-Enumerations of halved alternating sign matrices</a>, arXiv:math/0106038 [math.CO], 2001.
%H Theresia Eisenkölbl, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s46eisenko.html">2-Enumerations of halved alternating sign matrices</a>, Séminaire Lotharingien Combin. 46, (2001), Article B46c, 11 pp.
%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.
%H F. Harary and R. W. Robinson, <a href="http://dx.doi.org/10.4153/CJM-1979-007-3">Labeled bipartite blocks</a>, Canad. J. Math., 31 (1979), 60-68.
%H S. R. Kannan and Rajesh Kumar Mohapatra, <a href="https://arxiv.org/abs/1909.13678">Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques</a>, arXiv:1909.13678 [math.GM], 2019.
%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H Götz Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/01-Matrix.html">01-Matrix</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F G.f. satisfies: A(x) = 1 + 2*x*A(4x). - _Paul D. Hanna_, Dec 04 2009
%F a(n) = 2^n * Sum_{i = 0..C(n, 2)} C(C(n, 2), i)*3^i. The summation conditions on i, 0 <= i <= C(n, 2), the number of 1's above the main diagonal in the matrix representations of the relations on {1, 2, ..., n}. - _Geoffrey Critzer_, Feb 18 2011
%F G.f.: 1 / (1 - 2^1*x / (1 - 2^1*(2^2-1)*x / (1 - 2^5 * x / (1 - 2^3*(2^4-1)*x / (1 - 2^9*x / (1 - 2^5*(2^6-1)*x / ...)))))). - _Michael Somos_, May 12 2012
%F a(n) = [x^n] 1/(1 - 2^n*x). - _Ilya Gutkovskiy_, Oct 10 2017
%F Sum_{n>=0} 1/a(n) = A319015. - _Amiram Eldar_, Oct 14 2020
%e G.f. = 1 + 2*x + 16*x^2 + 512*x^3 + 65536*x^4 + 33554432*x^5 + ...
%t Table[2^(n^2), {n,0,15}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%o (PARI) a(n)=polresultant((x-1)^n,(x+1)^n,x) \\ _Ralf Stephan_
%o (PARI) a(n)=2^n^2 \\ _Charles R Greathouse IV_, Jun 23 2021
%o (Magma) [2^(n^2): n in [0..15]]; // _Vincenzo Librandi_, May 13 2011
%o (Sage) [2^(n^2) for n in (0..15)] # _G. C. Greubel_, Jul 03 2019
%o (GAP) List([0..15], n-> 2^(n^2) ) # _G. C. Greubel_, Jul 03 2019
%Y Bisection of A060656.
%Y Cf. A053763, A064062, A064231, A319015.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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