Displaying 1-10 of 88 results found.
1, 2, 3, 6, 12, 21, 42, 84, 147, 294
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), May 25 2003
Triangular array read by rows: T(n,k) = A002884(k)*2^((n-k)(n-k-1)), n >= 0, 0 <= k <= n.
+20
1
1, 1, 1, 4, 1, 6, 64, 4, 6, 168, 4096, 64, 24, 168, 20160, 1048576, 4096, 384, 672, 20160, 9999360, 1073741824, 1048576, 24576, 10752, 80640, 9999360, 20158709760, 4398046511104, 1073741824, 6291456, 688128, 1290240, 39997440, 20158709760, 163849992929280
COMMENTS
For A,B in the set of n X n matrices over GF(2) let A ~ B iff A^j = B^k for some positive j,k. Then ~ is an equivalence relation. There is exactly one idempotent matrix in each equivalence class. Let E be an idempotent matrix of rank k. Then T(n,k) is the size of the class containing E.
The classes in the equivalence relation described above are called the torsion classes corresponding to the idempotent E. - Geoffrey Critzer, Oct 02 2022
EXAMPLE
Triangle begins:
1;
1, 1;
4, 1, 6;
64, 4, 6, 168;
4096, 64, 24, 168, 20160;
1048576, 4096, 384, 672, 20160, 9999360;
...
T(3,1)=4 because we have: { I = {{0, 0, 0}, {0, 0, 0}, {0, 0, 1}},
A= {{0, 0, 0}, {1, 0, 0}, {0, 0, 1}}, B= {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}},
C= {{1, 1, 0}, {1, 1, 0}, {0, 0, 1}} } where I is idempotent of rank 1 and A^2=B^2=C^2=I.
MATHEMATICA
q = 2; nn = 7; Table[Table[Product[q^d - q^i, {i, 0, d - 1}] q^((n - d) (n - d - 1)), {d, 0, n}], {n, 0, nn}] // Grid
PROG
b(n) = {prod(i=2, n, 2^i-1)<<binomial(n, 2)}
Triangular array read by rows. T(n,k) = A002884(n)/ A002884(n-k)*2^((n-k)(n-k-1)), n>=0, 0<=k<=n.
+20
0
1, 1, 1, 4, 6, 6, 64, 112, 168, 168, 4096, 7680, 13440, 20160, 20160, 1048576, 2031616, 3809280, 6666240, 9999360, 9999360, 1073741824, 2113929216, 4095737856, 7679508480, 13439139840, 20158709760, 20158709760, 4398046511104, 8727373545472, 17182016667648, 33290157293568, 62419044925440, 109233328619520, 163849992929280, 163849992929280
COMMENTS
Let ~ be the equivalence relation on the set of n X n matrices over GF(2) defined by A ~ B if and only if the dimension of the image of A^n is equal to the dimension of the image of B^n. Let A be a recurrent matrix (Cf A348622) of rank k. Then T(n,k) is the size of the equivalence class containing A.
EXAMPLE
Triangle begins:
1,
1, 1,
4, 6, 6,
64, 112, 168, 168,
4096, 7680, 13440, 20160, 20160,
1048576, 2031616, 3809280, 6666240, 9999360, 9999360
MATHEMATICA
R[n_, d_] := Product[q^n - q^i, {i, 0, n - 1}]/Product[q^(n - d) - q^i, {i, 0, n - d - 1}]; Table[Table[R[n, d] q^((n - d) (n - d - 1)), {d, 0, n}], {n, 0, 10}] // Grid
a(n) = 2^(n*(n-1)/2).
(Formerly M1897)
+10
355
1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736, 35184372088832, 36028797018963968, 73786976294838206464, 302231454903657293676544, 2475880078570760549798248448, 40564819207303340847894502572032, 1329227995784915872903807060280344576
COMMENTS
Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments.
Number of perfect matchings of order n Aztec diamond. [see Speyer]
Number of Gelfand-Zeitlin patterns with bottom row [1,2,3,...,n]. [Zeilberger]
For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(2) (sequence A002884). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001 a(n) is the number of ways to tile the region
o-----o
|.....|
o--o.....o--o
|...........|
o--o...........o--o
|.................|
o--o.................o--o
|.......................|
|.......................|
|.......................|
o--o.................o--o
|.................|
o--o...........o--o
|...........|
o--o.....o--o
|.....|
o-----o
(top-to-bottom distance = 2n) with dominoes like either of
o--o o-----o
|..| or |.....|
|..| o-----o
|..|
o--o
(End)
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001 Let M_n denotes the n X n matrix with M_n(i,j)=binomial(2i,j); then det(M_n)=a(n+1). - Benoit Cloitre, Apr 21 2002 Smallest power of 2 which can be expressed as the product of n distinct numbers (powers of 2), e.g., a(4) = 1024 = 2*4*8*16. Also smallest number which can be expressed as the product of n distinct powers. - Amarnath Murthy, Nov 10 2002 The number of binary relations that are both reflexive and symmetric on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
The number of symmetric binary relations on an (n-1)-element set. - Peter Kagey, Feb 13 2021 To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125. The probability of having won before n+1 tails is A114604 / A006125. - Joshua Zucker, Dec 14 2005 a(n-1) is the number of simple labeled graphs on n nodes such that every node has even degree. - Geoffrey Critzer, Oct 21 2011 a(n+1) is the number of symmetric binary matrices of size n X n. - Nathan J. Russell, Aug 30 2014 Let T_n be the n X n matrix with T_n(i,j) = binomial(2i + j - 3, j-1); then det(T_n) = a(n). - Tony Foster III, Aug 30 2018 k^(n*(n-1)/2) is the determinant of n X n matrix T_(i,j) = binomial(k*i + j - 3, j-1), in this case k=2. - Tony Foster III, May 12 2019 Let B_n be the n+1 X n+1 matrix with B_n(i, j) = Sum_{m=max(0, j-i)..min(j, n-i)} (binomial(i, j-m) * binomial(n-i, m) * (-1)^m), 0<=i,j<=n. Then det B_n = a(n+1). Also, deleting the first row and any column from B_n results in a matrix with determinant a(n). The matrices B_n have the following property: B_n * [x^n, x^(n-1) * y, x^(n-2) * y^2, ..., y^n]^T = [(x-y)^n, (x-y)^(n-1) * (x+y), (x-y)^(n-2) * (x+y)^2, ..., (x+y)^n]^T. - Nicolas Nagel, Jul 02 2019 a(n) is the number of positive definite (-1,1)-matrices of size n X n. - Eric W. Weisstein, Jan 03 2021 a(n) is the number of binary relations on a labeled n-set that are both total and antisymmetric. - José E. Solsona, Feb 05 2023
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 547 (Fig. 9.7), 573.
G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2).
J. Propp, Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics, L. Billera et al., eds., Mathematical Sciences Research Institute series, vol. 38, Cambridge University Press, 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
FORMULA
Sequence is given by the Hankel transform of A001003 (Schroeder's numbers) = 1, 1, 3, 11, 45, 197, 903, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - Philippe Deléham, Mar 02 2004 a(n) = 2^floor(n^2/2)/2^floor(n/2). - Paul Barry, Oct 04 2004 G.f.: G(0)/x - 1/x, where G(k) = 1 + 2^(k-1)*x/(1 - 1/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013 a(n) = s_lambda(1,1,...,1) where s is the Schur polynomial in n variables and lambda is the partition (n,n-1,n-2,...,1). - Leonid Bedratyuk, Feb 06 2022 a(n) = Product_{1 <= j <= i <= n-1} (i + j)/(2*i - 2*j + 1). Cf. A007685. - Peter Bala, Oct 25 2024
EXAMPLE
This sequence counts labeled graphs on n vertices. For example, the a(0) = 1 through a(2) = 8 graph edge sets are:
{} {} {} {}
{12} {12}
{13}
{23}
{12,13}
{12,23}
{13,23}
{12,13,23}
This sequence also counts labeled graphs with loops on n - 1 vertices. For example, the a(1) = 1 through a(3) = 8 edge sets are the following. A loop is represented as an edge with two equal vertices.
{} {} {}
{11} {11}
{12}
{22}
{11,12}
{11,22}
{12,22}
{11,12,22}
(End)
MATHEMATICA
Join[{1}, 2^Accumulate[Range[0, 20]]] (* Harvey P. Dale, May 09 2013 *) Prepend[Table[Count[Tuples[{0, 1}, {n, n}], _?SymmetricMatrixQ], {n, 5}], 1] (* Eric W. Weisstein, Jan 03 2021 *) Prepend[Table[Count[Tuples[{-1, 1}, {n, n}], _?PositiveDefiniteMatrixQ], 1], {n, 4}] (* Eric W. Weisstein, Jan 03 2021 *)
PROG
(Haskell) [2^(n*(n-1) `div` 2) | n <- [0..20]] -- José E. Solsona, Feb 05 2023 (Python)
CROSSREFS
Cf. A001187 (connected labeled graphs). The unlabeled version is A000088, or A002494 without isolated vertices. The version for hypergraphs is A058891, or A016031 without singletons. The case of connected edge set is A287689.
Decimal expansion of Product_{k >= 1} (1 - 1/2^k).
+10
99
2, 8, 8, 7, 8, 8, 0, 9, 5, 0, 8, 6, 6, 0, 2, 4, 2, 1, 2, 7, 8, 8, 9, 9, 7, 2, 1, 9, 2, 9, 2, 3, 0, 7, 8, 0, 0, 8, 8, 9, 1, 1, 9, 0, 4, 8, 4, 0, 6, 8, 5, 7, 8, 4, 1, 1, 4, 7, 4, 1, 0, 6, 6, 1, 8, 4, 9, 0, 2, 2, 4, 0, 9, 0, 6, 8, 4, 7, 0, 1, 2, 5, 7, 0, 2, 4, 2, 8, 4, 3, 1, 9, 3, 3, 4, 8, 0, 7, 8, 2
COMMENTS
This is the limiting probability that a large random binary matrix is nonsingular (cf. A002884). This constant is very close to 2^(13/24) * sqrt(Pi/log(2)) / exp(Pi^2/(6*log(2))) = 0.288788095086602421278899775042039398383022429351580356839... - Vaclav Kotesovec, Aug 21 2018
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
FORMULA
Lim inf Product_{k=0..floor(log_2(n))} floor(n/2^k)*2^k/n for n->oo. - Hieronymus Fischer, Aug 13 2007 Lim inf A098844(n)/n^(1+floor(log_2(n)))*2^(1/2*(1+floor(log_2(n)))*floor(log_2(n))) for n->oo. - Hieronymus Fischer, Aug 13 2007 Product_{k >= 1} (1-1/2^k) = (1/2; 1/2)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015 exp(Sum_{n>=1}(1/n/(1 - 2^n))) (according to Mathematica). - Mats Granvik, Sep 07 2016 (Sum_{k>0} (4^k-1)/(Product_{i=1..k} ((4^i-1)*(2*4^i-1))))*2 = 2/7 + 2/(3*7*31) + 2/(3*7*15*31*127)+2/(3*7*15*31*63*127*511) + ... (conjecture). - Werner Schulte, Dec 22 2016 Equals Sum_{k=-oo..oo} (-1)^k/2^((3*k+1)*k/2) (by Euler's pentagonal number theorem). - Amiram Eldar, Aug 13 2020 Constant C = Sum_{n >= 0} (-1)^n/( Product_{k = 1..n} (2^k - 1) ). The above conjectural result by Schulte follows by adding terms of this series in pairs.
C = (1/2)*Sum_{n >= 0} (-1/2)^n/( Product_{k = 1..n} (2^k - 1) ).
C = (3/8)*Sum_{n >= 0} (-1/4)^n/( Product_{k = 1..n} (2^k - 1) ).
1/C = Sum_{n >= 0} 2^(n*(n-1)/2)/( Product_{k = 1..n} (2^k - 1) ).
C = 1 - Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 - 1/2^k).
This latter identity generalizes as:
C = Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*C = 1 - Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*C = 6 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*15*C = 91 - Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
and so on, where the sequence [1, 0, 1, 6, 91, ...] is A005327. (End)
Equals sqrt(2*Pi/log(2)) * exp(log(2)/24 - Pi^2/(6*log(2))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/ A005329(n).
EXAMPLE
(1/2)*(3/4)*(7/8)*(15/16)*... = 0.288788095086602421278899721929230780088911904840685784114741...
MATHEMATICA
RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
PROG
(PARI) default(realprecision, 20080); x=prodinf(k=1, -1/2^k, 1); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b048651.txt", n, " ", d)); \\ Harry J. Smith, May 07 2009
CROSSREFS
Cf. A002884, A001318, A005327, A005329, A048652, A065446, A079555, A098844, A067080, A100220, A132019, A132020, A132026, A132038, A070933, A261584, A335764.
1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 4722366482869645213696, 1237940039285380274899124224, 1298074214633706907132624082305024, 5444517870735015415413993718908291383296, 91343852333181432387730302044767688728495783936
COMMENTS
Nilpotent n X n matrices over GF(2). Also number of simple digraphs (without self-loops) on n labeled nodes (see also A002416). For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(4) (sequence A053291). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001 (-1)^ceiling(n/2) * resultant of the Chebyshev polynomial of first kind of degree n and Chebyshev polynomial of first kind of degree (n+1) (cf. A039991). - Benoit Cloitre, Jan 26 2003 The number of reflexive binary relations on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
Number of gift exchange scenarios where, for each person k of n people,
i) k gives gifts to g(k) of the others, where 0 <= g(k) <= n-1,
ii) k gives no more than one gift to any specific person,
iii) k gives no single gift to two or more people and
iv) there is no other person j such that j and k jointly give a single gift.
(In other words -- but less precisely -- each person k either gives no gifts or gives exactly one gift per person to 1 <= g(k) <= n-1 others.) (End)
In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 4 types of edge. To clarify the comment from Benoit Cloitre, dated Jan 26 2003, in this context: simple digraphs (without self-loops) have four types of edge. These types of edges are as follows: the absent edge, the directed edge from A -> B, the directed edge from B -> A and the bidirectional edge, A <-> B. - Mark Stander, Apr 11 2019
REFERENCES
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 521.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 5, Eq. (1.1.5).
FORMULA
Sequence given by the Hankel transform (see A001906 for definition) of A059231 = {1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, ...}; example: det([1, 1, 5, 29; 1, 5, 29, 185; 5, 29, 185, 1257; 29, 185, 1257, 8925]) = 4^6 = 4096. - Philippe Deléham, Aug 20 2005
EXAMPLE
a(2)=4 because there are four 2 x 2 nilpotent matrices over GF(2):{{0,0},{0,0}},{{0,1},{0,0}},{{0,0},{1,0}},{{1,1,},{1,1}} where 1+1=0. - Geoffrey Critzer, Oct 05 2012
MAPLE
a:=n->mul(4^j, j=1..n-1): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 03 2007
a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).
+10
64
1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128
COMMENTS
Informally, write down 1 followed by 2^k 2^(k-1) times, for k = 1,2,3,4,... These are the denominators of the binary van der Corput sequence (see A030101 for the numerators). - N. J. A. Sloane, Dec 01 2019 a(n) is the denominator of the form 2^k needed to make the ratio (2n-1)/2^k lie in the interval [1-2], i.e. such ratios are 1/1, 3/2, 5/4, 7/4, 9/8, 11/8, 13/8, 15/8, 17/16, 19/16, 21/16, ... where the numerators are A005408 (The odd numbers). Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the diagonal and 1 elsewhere. For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /. The order of this matrix as an element of GL(n,2) is a(n-1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001
A006257(n)/a(n) = (0, 0.1, 0.01, 0.11, 0.001, ...) enumerates all binary fractions in the unit interval [0, 1). - Fredrik Johansson, Aug 14 2006
This is the discriminator sequence for the odious numbers. - N. J. A. Sloane, May 10 2016
FORMULA
a(1) = 1 and a(n+1) = a(n)*ceiling(n/a(n)). - Benoit Cloitre, Aug 17 2002 G.f.: 1/(1-x) * (1 + Sum_{k>=0} 2^k*x^2^k). - Ralf Stephan, Apr 18 2003 a(n) is the smallest power of 2 > n. - Chai Wah Wu, Nov 04 2016
MAPLE
[seq(2^(floor_log_2(j)+1), j=0..127)]; or [seq(coerce1st_octave((2*j)+1), j=0..127)]; or [seq(a(j), j=0..127)];
coerce1st_octave := proc(r) option remember; if(r < 1) then coerce1st_octave(2*r); else if(r >= 2) then coerce1st_octave(r/2); else (r); fi; fi; end;
option remember;
if n = 0 then
1 ;
else
2*procname(floor(n/2));
end if;
end proc:
A062383 := n -> 1 + Bits:-Iff(n, n):
MATHEMATICA
a[n_] := a[n] = 2 a[n/2 // Floor]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2016 *)
PROG
(PARI) { a=1; for (n=0, 1000, write("b062383.txt", n, " ", a*=ceil((n + 1)/a)) ) } \\ Harry J. Smith, Aug 06 2009 (Haskell)
import Data.List (transpose)
a062383 n = a062383_list !! n
a062383_list = 1 : zs where
zs = 2 : (map (* 2) $ concat $ transpose [zs, zs])
(Magma) [2^Floor(Log(2, 2*n+1)): n in [0..70]]; // Bruno Berselli, Mar 04 2016 (Python)
Number of nonsingular n X n matrices over GF(3).
+10
30
1, 2, 48, 11232, 24261120, 475566474240, 84129611558952960, 134068444202678083338240, 1923442429811445711790394572800, 248381049201184165590947520186915225600, 288678833735376059528974260112416365258106470400
FORMULA
a(n) = Product_{k=0..n-1}(3^n-3^k). - corrected by Michel Marcus, Sep 18 2015
PROG
(Magma) [1] cat [&*[(3^n - 3^k): k in [0..n-1]]: n in [1..9]]; // Bruno Berselli, Jan 28 2013 (PARI) for(n=0, 10, print1(prod(k=0, n-1, 3^n - 3^k), ", ")) \\ G. C. Greubel, May 31 2018
Decimal expansion of Product_{k>=1} (1-1/2^k)^(-1).
+10
25
3, 4, 6, 2, 7, 4, 6, 6, 1, 9, 4, 5, 5, 0, 6, 3, 6, 1, 1, 5, 3, 7, 9, 5, 7, 3, 4, 2, 9, 2, 4, 4, 3, 1, 1, 6, 4, 5, 4, 0, 7, 5, 7, 9, 0, 2, 9, 0, 4, 4, 3, 8, 3, 9, 1, 3, 2, 9, 3, 5, 3, 0, 3, 1, 7, 5, 8, 9, 1, 5, 4, 3, 9, 7, 4, 0, 4, 2, 0, 6, 4, 5, 6, 8, 7, 9, 2, 7, 7, 4, 0, 2, 9, 4, 8, 4, 3, 3, 5, 3, 5, 0, 8, 8, 0
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
FORMULA
Equals 1/QPochhammer(1/2, 1/2)_{infinity}. - G. C. Greubel, Jan 18 2018 Equals 1 + Sum_{n>=1} 2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). - Robert FERREOL, Feb 22 2020
EXAMPLE
3.46274661945506361153795734292443116454075790290...
MAPLE
evalf(1+sum(2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 22 2020
MATHEMATICA
N[ Product[ 1/(1 - 1/2^k), {k, 1, Infinity} ], 500 ]
RealDigits[1/QPochhammer[1/2, 1/2], 10, 100][[1]] (* Vaclav Kotesovec, Jun 22 2014 *)
PROG
(PARI) { default(realprecision, 2080); x=prodinf(k=1, 1/(1 - 1/2^k)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065446.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009
Number of invertible n X n matrices with entries equal to 0 or 1.
+10
20
1, 1, 6, 174, 22560, 12514320, 28836612000, 270345669985440, 10160459763342013440
COMMENTS
All eigenvalues are nonzero.
EXAMPLE
For n=2 the 6 matrices are {{{0, 1}, {1, 0}}, {{0, 1}, {1, 1}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}, {{1, 1}, {1, 0}}}.
PROG
(PARI) a(n)=sum(t=0, 2^n^2-1, !!matdet(matrix(n, n, i, j, (t>>(i*n+j-n-1))%2))) \\ Charles R Greathouse IV, Feb 09 2016 (Python)
from itertools import product
from sympy import Matrix
def A055165(n): return sum(1 for s in product([0, 1], repeat=n**2) if Matrix(n, n, s).det() != 0) # Chai Wah Wu, Sep 24 2021
CROSSREFS
A046747(n) + a(n) = 2^(n^2) = total number of n X n (0, 1) matrices = sequence A002416.
AUTHOR
Ulrich Hermisson (uhermiss(AT)server1.rz.uni-leipzig.de), Jun 18 2000
EXTENSIONS
More terms from Miodrag Zivkovic (ezivkovm(AT)matf.bg.ac.rs), Feb 28 2006
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