A053291 -id:A053291 - cp's OEIS frontend

A053763 a(n) = 2^(n^2 - n).

1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 4722366482869645213696, 1237940039285380274899124224, 1298074214633706907132624082305024, 5444517870735015415413993718908291383296, 91343852333181432387730302044767688728495783936

Offset: 0

Stephen G Penrice, Mar 29 2000

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
From Rick L. Shepherd, Dec 24 2008: (Start)
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

			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

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).

T. D. Noe, Table of n, a(n) for n = 0..35
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
Murray Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., Vol. 5 (1961), 330-333.
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.
Greg Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Annals of mathematics, Second Series, Vol. 156, No. 3 (2002), pp. 835-866, arXiv preprint, arXiv:math/0008184 [math.CO], 2000-2001 (see Th. 3).
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.

Row sums of A123554, A189898, A346412, A346214.

Cf. A053773, A006125, A000273, A000984, A002416, A319016.

seq(4^(binomial(n, n-2)), n=0..12); # Zerinvary Lajos, Jun 16 2007
a:=n->mul(4^j, j=1..n-1): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 03 2007

Table[2^(2*Binomial[n,2]), {n,0,20}] (* Geoffrey Critzer, Oct 04 2012 *)

a(n)=1<<(n^2-n) \\ Charles R Greathouse IV, Nov 20 2012

def A053763(n): return 1<Chai Wah Wu, Jul 05 2024

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
a(n) = 4^binomial(n, n-2). - Zerinvary Lajos, Jun 16 2007
a(n) = Sum_{i=0..n^2-n} binomial(n^2-n, i). - Rick L. Shepherd, Dec 24 2008
G.f. A(x) satisfies: A(x) = 1 + x * A(4*x). - Ilya Gutkovskiy, Jun 04 2020
Sum_{n>=1} 1/a(n) = A319016. - Amiram Eldar, Oct 27 2020
Sum_{n>=0} a(n)*u^n/A002884(n) = Product_{r>=1} 1/(1-u/q^r). - Geoffrey Critzer, Oct 28 2021

A053290 Number of nonsingular n X n matrices over GF(3).

Original entry on oeis.org

1, 2, 48, 11232, 24261120, 475566474240, 84129611558952960, 134068444202678083338240, 1923442429811445711790394572800, 248381049201184165590947520186915225600, 288678833735376059528974260112416365258106470400

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..45
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=3 of A316622.
Cf. A002884, A003787, A027871, A047656, A053291, A053292, A053293.
Cf. A100220, A219205.

[1] cat [&*[(3^n - 3^k): k in [0..n-1]]: n in [1..9]]; // Bruno Berselli, Jan 28 2013

Table[Product[3^n - 3^k, {k, 0, n - 1}], {n, 0, 10}] (* Geoffrey Critzer, Jan 26 2013; edited by Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 3^n - 3^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = Product_{k=0..n-1}(3^n-3^k). - corrected by Michel Marcus, Sep 18 2015
a(n) = A047656(n)*A027871(n). - Bruno Berselli, Jan 30 2013
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} A219205(k).
a(n) ~ c * 3^(n^2), where c = A100220. (End)

More terms from Vladeta Jovovic, Mar 16 2000

A053292 Number of nonsingular n X n matrices over GF(5).

Original entry on oeis.org

1, 4, 480, 1488000, 116064000000, 226614960000000000, 11064475422000000000000000, 13506266841692625000000000000000000, 412177498341354683437500000000000000000000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..35
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=5 of A316622.

Cf. A002884, A003789, A027872, A053290, A053291, A053293, A109345, A100222.

[1] cat [&*[(5^n - 5^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[5^n - 5^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 5^n - 5^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (5^n - 1)*(5^n - 5)*...*(5^n - 5^(n-1)).
a(n) = A109345(n)*A027872(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 5^(n^2), where c = A100222. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A053293 Number of nonsingular n X n matrices over GF(7).

Original entry on oeis.org

1, 6, 2016, 33784128, 27811094169600, 1122211189922928537600, 2218959336124989671614429593600, 214992513152176999576908105619651923148800, 1020690003311610463765638355505358381593396977336320000, 237443634207909205360438080389756681126654524500073656592021585920000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=7 of A316622.

Cf. A002884, A003790, A027875, A053290, A053291, A053292, A109493, A132035.

[1] cat [&*[(7^n - 7^k): k in [0..n-1]]: n in [1..7]]; // Bruno Berselli, Jan 28 2013

Table[Product[7^n - 7^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 7^n - 7^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (7^n - 1)*(7^n - 7)*...*(7^n - 7^(n-1)).
a(n) = A109493(n)*A027875(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 7^(n^2), where c = A132035. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 2, 48, 168, 1, 1, 4, 96, 11232, 20160, 1, 1, 2, 480, 86016, 24261120, 9999360, 1, 1, 6, 288, 1488000, 1321205760, 475566474240, 20158709760, 1, 1, 4, 2016, 1886976, 116064000000, 335522845163520, 84129611558952960, 163849992929280, 1

Offset: 0

Andrew Howroyd, Jul 08 2018

All rows are multiplicative.
Equivalently, the number of invertible n X n matrices mod k.
Also, for k prime (but not higher prime powers) the number of nonsingular n X n matrices over GF(k).
For k >= 2, n! divides T(n,k) since the subgroup of GL(n,k) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). Note that a permutation matrix is an orthogonal matrix, hence having determinant +-1. - Jianing Song, Oct 29 2022

			Array begins:
=================================================================
n\k| 1       2         3          4             5           6
---+-------------------------------------------------------------
0  | 1       1         1          1            1            1 ...
1  | 1       1         2          2            4            2 ...
2  | 1       6        48         96          480          288 ...
3  | 1     168     11232      86016      1488000      1886976 ...
4  | 1   20160  24261120 1321205760 116064000000 489104179200 ...
5  | 1 9999360  ...
...

R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Mathematics 66 (1987) pp. 35-49.
J. M. Lockhart and W. P. Wardlaw, Determinants of Matrices over the Integers Modulo m, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.
The Group Properties Wiki, Order formulas for linear groups

Rows n=2..4 are A000252, A064767, A305186.
Columns k=2..7 are A002884, A053290, A065128, A053292, A065498, A053293.
Cf. A053291 (GF(4)), A052496 (GF(8)), A052497 (GF(9)).
Cf. A316623.

T:=function(n,k) if k=1 or n=0 then return 1; else return Order(GL(n, Integers mod k)); fi; end;
for n in [0..5] do Print(List([1..6], k->T(n,k)), "\n"); od;

T[, 1] = T[0, ] = 1; T[n_, k_] := T[n, k] = Module[{f = FactorInteger[k], p, e}, If[Length[f] == 1, {p, e} = f[[1]]; (p^e)^(n^2)* Product[(1 - 1/p^j), {j, 1, n}], Times @@ (T[n, Power @@ #]& /@ f)]];
Table[T[n - k + 1, k], {n, 0, 8}, {k, n + 1, 1, -1}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

T(n,k)={my(f=factor(k)); k^(n^2) * prod(i=1, #f~, my(p=f[i,1]); prod(j=1, n, (1 - p^(-j))))}

T(n,p^e) = (p^e)^(n^2) * Product_{j=1..n} (1 - 1/p^j) for prime p.

A052496 Number of nonsingular n X n matrices over GF(8).

Original entry on oeis.org

1, 7, 3528, 115379712, 241909719367680, 32467582052437076213760, 278893342293098904613804037898240, 153323163270070838469523866093442017326530560

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30

Cf. A027876, A109966.

Cf. A002884, A003791, A053290, A053291, A053292, A053293, A052497, A052498, A132036.

[1] cat [&*[(8^n-8^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 30 2013

Table[Product[(8^n - 8^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 8^n - 8^j)}; \\ G. C. Greubel, May 14 2019

[product(8^n - 8^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (8^n - 1)*(8^n - 8)*...*(8^n - 8^(n-1)).
a(n) = A109966(n)*A027876(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 8^(n^2), where c = A132036. - Amiram Eldar, Jul 06 2025

A052497 Number of nonsingular n X n matrices over GF(9).

Original entry on oeis.org

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025

A003788 Order of universal Chevalley group A_n (4).

Original entry on oeis.org

1, 60, 60480, 987033600, 258492255436800, 1083930404878024704000, 72736898347485916060188672000, 78099458182389588115529148326215680000, 1341733356588640095264385107865053233298800640000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Index entries for sequences related to groups.

Cf. A003787, A003789, A003790, A003791, A003792, A053291, A100221.

[&*[(4^n - 4^k): k in [0..n-1]]/3: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[4, #] & /@ Range[0, 8] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053291(n)/3. - Ralf Stephan, Mar 30 2004
a(n) = A(4,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 4^(n*(n+2)), where c = (4/3) * A100221 = 0.918050049493... . - Amiram Eldar, Jul 07 2025

One more term from Sean A. Irvine, Sep 18 2015

A052498 Number of nonsingular n X n matrices over GF(11).

Original entry on oeis.org

1, 10, 13200, 2124276000, 41393302251840000, 97602635428252959312000000, 27847155251069188894843979022720000000, 961359275427083998992553051820498439890246400000000

Offset: 0

Vladeta Jovovic, Mar 16 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..30

Cf. A027879, A110195.

Cf. A002884, A053290, A053291, A053292, A053293, A052496, A052497, A132267.

[1] cat [&*[(11^n - 11^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[11^n - 11^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)

{a(n) = prod(j=0,n-1, 11^n - 11^j)}; \\ G. C. Greubel, May 14 2019

[product(11^n - 11^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (11^n - 1)*(11^n - 11)*...*(11^n - 11^(n-1)).
a(n) = A110195(n)*A027879(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 11^(n^2), where c = A132267. - Amiram Eldar, Jul 06 2025

A065128 Number of invertible n X n matrices mod 4 (i.e., over the ring Z_4).

Original entry on oeis.org

1, 2, 96, 86016, 1321205760, 335522845163520, 1385295986380096143360, 92239345887620476544860815360, 98654363640526679389774053813465907200, 1691558770638735027870457216848672340872423014400, 464518059995994038184379206447729320401459864818351813427200

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 14 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=4 of A316622.

Cf. A048651, A053291, A060195.

a[n_] := 4^(n^2)*Product[1 - 1/2^k, {k, 1, n} ]; Table[ a[n], {n, 0, 10} ]

for(n=1,11,print(4^(n^2)*prod(k=1,n,(1-1/2^k))))

a(n) = 4^(n^2) * Product_{k=1..n} (1 - 1/2^k).
a(n) = 2^(n^2) * A002884(n). - Geoffrey Critzer, Feb 04 2018
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} 2*A060195(k).
a(n) ~ c * 4^(n^2), where c = A048651. (End)

More terms from Robert G. Wilson v and Jason Earls, Nov 16 2001

cp's OEIS Frontend

A053763 a(n) = 2^(n^2 - n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A053290 Number of nonsingular n X n matrices over GF(3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A053292 Number of nonsingular n X n matrices over GF(5).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A053293 Number of nonsingular n X n matrices over GF(7).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

A052496 Number of nonsingular n X n matrices over GF(8).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A052497 Number of nonsingular n X n matrices over GF(9).

Views

Author