A002416 a(n) = 2^(n^2).

1, 2, 16, 512, 65536, 33554432, 68719476736, 562949953421312, 18446744073709551616, 2417851639229258349412352, 1267650600228229401496703205376, 2658455991569831745807614120560689152, 22300745198530623141535718272648361505980416, 748288838313422294120286634350736906063837462003712

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

Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Theresia Eisenkölbl, 2-Enumerations of halved alternating sign matrices, arXiv:math/0106038 [math.CO], 2001.

Theresia 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 of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.

F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv:1909.13678 [math.GM], 2019.

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.

Index to divisibility sequences

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

Sum_{n>=0} 1/a(n) = A319015. - Amiram Eldar, Oct 14 2020

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
(PARI) a(n)=2^n^2 \\ Charles R Greathouse IV, Jun 23 2021
(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. A053763, A064062, A064231, A319015.

Sequence in context: A293150 A286039 A063391 * A013028 A136632 A168405

Adjacent sequences: A002413 A002414 A002415 * A002417 A002418 A002419

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved