



1, 2, 16, 512, 65536, 33554432, 68719476736, 562949953421312, 18446744073709551616, 2417851639229258349412352, 1267650600228229401496703205376, 2658455991569831745807614120560689152, 22300745198530623141535718272648361505980416, 748288838313422294120286634350736906063837462003712
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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 selfloops (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 nset.  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


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. Eisenkoelbl, 2Enumerations of halved alternating sign matrices.
T. Eisenk\"olbl, 2Enumerations of halved alternating sign matrices
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), 6068.
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, 01Matrix
Index to divisibility sequences


FORMULA

G.f. satisfies: A(x) = 1 + 2*x*A(4x).  Paul D. Hanna, Dec 04 2009
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*(2^21)*x / (1  2^5 * x / (1  2^3*(2^41)*x / (1  2^9*x / (1  2^5*(2^61)*x / ...)))))).  Michael Somos, May 12 2012


EXAMPLE

1 + 2*x + 16*x^2 + 512*x^3 + 65536*x^4 + 33554432*x^5 + ...


MATHEMATICA

Table[2^(n^2), {n, 0, 9}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)


PROG

(PARI) a(n)=polresultant((x1)^n, (x+1)^n, x) \\ Ralf Stephan
(MAGMA) [2^(n^2): n in [0..15]]; // Vincenzo Librandi, May 13 2011


CROSSREFS

Bisection of A060656. Cf. also A064231, A053763.
Sequence in context: A063387 A228979 A063391 * A013028 A136632 A168405
Adjacent sequences: A002413 A002414 A002415 * A002417 A002418 A002419


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



