A064231 -id:A064231 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

John M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). - Abdullahi Umar, Sep 14 2008

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

Bisection of A060656.

Cf. A053763, A064062, A064231, A319015.

List([0..15], n-> 2^(n^2) ); # G. C. Greubel, Jul 03 2019

[2^(n^2): n in [0..15]]; // Vincenzo Librandi, May 13 2011

Table[2^(n^2), {n,0,15}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)

a(n)=polresultant((x-1)^n,(x+1)^n,x) \\ Ralf Stephan

a(n)=2^n^2 \\ Charles R Greathouse IV, Jun 23 2021

[2^(n^2) for n in (0..15)] # G. C. Greubel, Jul 03 2019

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 the number of symmetric pairs (a,b) with aA027465, A013610. - Geoffrey Critzer, Nov 05 2024

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

A000410 Number of singular n X n rational (0,1)-matrices.

Original entry on oeis.org

0, 0, 6, 425, 65625, 27894671, 35716401889, 144866174953833

Offset: 1

N. J. A. Sloane

Number of all n X n (0,1)-matrices having distinct, nonzero ordered rows and determinant 0 - compare A000409.

a(n) is the number of singular n X n rational {0,1}-matrices with no zero rows and with all rows distinct, up to permutation of rows and so a(n) = binomial(2^n-1,n) - A088389(n). Cf. A116506, A116507, A116527, A116532. - Vladeta Jovovic, Apr 03 2006

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Metropolis and P. R. Stein, On a class of (0,1) matrices with vanishing determinants, J. Combin. Theory, 3 (1967), 191-198.
Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.
Index entries for sequences related to binary matrices

Cf. A000409, A046747, A064230, A064231.

n! * a(n) = A046747(n) - 2^(n^2) + n! * binomial(2^n -1, n).

n=7 term from Guenter M. Ziegler (ziegler(AT)math.TU-Berlin.DE)

a(8) from Vladeta Jovovic, Mar 28 2006

A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 9, 6, 1, 49, 288, 174, 1, 225, 6750, 36000, 22560, 1, 961, 118800, 3159750, 17760600, 12514320, 1, 3969, 1807806, 190071000, 5295204600, 34395777360, 28836612000, 1, 16129, 25316928, 9271660734, 1001080231200, 32307576315840

Offset: 0

N. J. A. Sloane, Sep 23 2001

Rows add to 2^(n^2).
Komlos and later Kahn, Komlos and Szemeredi show that almost all such matrices are invertible.
Table 3 from M. Zivkovic, Classification of small (0,1) matrices (see link). - Vladeta Jovovic, Mar 28 2006

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   9,      6;
  1,  49,    288,     174;
  1, 225,   6750,   36000,    22560;
  1, 961, 118800, 3159750, 17760600, 12514320;
  ...

J. Kahn, J. Komlos and E. Szemeredi: On the probability that a random +-1 matrix is singular, J. AMS 8 (1995), 223-240.
J. Komlos, On the determinants of random matrices, Studia Sci. Math. Hungar., 3 (1968), 387-399.

M. Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.

Cf. A064231, A000409, A000410, A046747, A086875.

Main diagonal gives A055165.

T=matrix(5,5); { for(n=0,4, mm=matrix(n,n); for(k=0,n,T[1+n,1+k]=0); forvec(x=vector(n*n,i,[0,1]), for(i=1,n, for(j=1,n,mm[i,j]=x[i+n*(j-1)])); T[1+n,1+matrank(mm)]++); for(k=0,n,print1(T[1+n,1+k], if(k

Sum_{k=1..n} k * T(n,k) = A086875(n). - Alois P. Heinz, Jun 18 2022

More terms and PARI code from Michael Somos, Sep 25 2001
6 more terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 17 2004
More terms from Vladeta Jovovic, Mar 28 2006

A173760 Partials sums of A000410.

Original entry on oeis.org

0, 0, 6, 431, 66056, 27960727, 35744362616, 144901919316449

Offset: 1

Jonathan Vos Post, Feb 23 2010

Partials sums of number of singular n X n rational (0,1)-matrices. The subsequence of primes in this partial sum begins: 431.

			a(8) = 0 + 0 + 6 + 425 + 65625 + 27894671 + 35716401889 + 144866174953833.

Cf. A000409, A000410, A046747, A064230, A064231.

a(n) = SUM[i=0..n] A000410(i).

cp's OEIS Frontend

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

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A000410 Number of singular n X n rational (0,1)-matrices.

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A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n).

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A173760 Partials sums of A000410.

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