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 A002416 a(n) = 2^(n^2). 77
 1, 2, 16, 512, 65536, 33554432, 68719476736, 562949953421312, 18446744073709551616, 2417851639229258349412352, 1267650600228229401496703205376, 2658455991569831745807614120560689152, 22300745198530623141535718272648361505980416, 748288838313422294120286634350736906063837462003712 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For n >= 1 a(n) is the number of n X n (0, 1) matrices. Also number of directed graphs on n labeled nodes allowing self-loops (cf. A053763). 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 Hankel transform of A064062. - Philippe Deléham, Nov 19 2007 a(n) is also the order of the semigroup (monoid) of all binary relations on an n-set. - Abdullahi Umar, Sep 14 2008 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 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 REFERENCES John M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). - Abdullahi Umar, Sep 14 2008 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..33 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. T. Eisenkölbl, 2-Enumerations of halved alternating sign matrices, arXiv:math/0106038 [math.CO], 2001. T. Eisenkölbl, 2-Enumerations of halved alternating sign matrices, Séminaire Lotharingien Combin. 46, (2001), Article B46c, 11 pp. Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003. F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1. Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2. Eric Weisstein's World of Mathematics, 01-Matrix FORMULA G.f. satisfies: A(x) = 1 + 2*x*A(4x). - Paul D. Hanna, Dec 04 2009 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 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 a(n) = [x^n] 1/(1 - 2^n*x). - Ilya Gutkovskiy, Oct 10 2017 EXAMPLE G.f. = 1 + 2*x + 16*x^2 + 512*x^3 + 65536*x^4 + 33554432*x^5 + ... MATHEMATICA Table[2^(n^2), {n, 0, 15}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *) PROG (PARI) a(n)=polresultant((x-1)^n, (x+1)^n, x) \\ Ralf Stephan (MAGMA) [2^(n^2): n in [0..15]]; // Vincenzo Librandi, May 13 2011 (Sage) [2^(n^2) for n in (0..15)] # G. C. Greubel, Jul 03 2019 (GAP) List([0..15], n-> 2^(n^2) ) # G. C. Greubel, Jul 03 2019 CROSSREFS Bisection of A060656. Cf. also A064231, A053763. Sequence in context: A293150 A286039 A063391 * A013028 A136632 A168405 Adjacent sequences:  A002413 A002414 A002415 * A002417 A002418 A002419 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)