A053601 -id:A053601 - cp's OEIS frontend

A028365 Order of general affine group over GF(2), AGL(n,2).

1, 2, 24, 1344, 322560, 319979520, 1290157424640, 20972799094947840, 1369104324918194995200, 358201502736997192984166400, 375234700595146883504949480652800, 1573079924978208093254925489963584716800

Offset: 0

N. J. A. Sloane

For n > 0, a(n) = v(n+1)/v(n), where v = A203305 is the Vandermonde determinant of the first n of the numbers -2^j - 1; see the Mathematica section. - Clark Kimberling, Jan 01 2012

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 54 (1.64).

Seiichi Manyama, Table of n, a(n) for n = 0..57
Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, SZ Sequences: Binary-Based (0, 2^q)-Sequences, arXiv:2505.20434 [cs.GR], 2025. See p. 7.
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Putnam Competition 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.

Cf. A000217, A002884, A005329, A020522, A048651, A053601, A203305.

[1] cat [(&*[2^(n+1) - 2^(j+1): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 31 2023

A028365 := n->2^n*product(2^n-2^'i','i'=0..n-1); # version 1
A028365 := n->product(2^'j'-1,'j'=1..n)*2^binomial(n+1,2); # version 2

RecurrenceTable[{a[1]==1, a[2]==2, a[3]==24, a[n]==(6a[n-1]^2 a[n-3] - 8a[n-1] a[n-2]^2)/(a[n-2] a[n-3])}, a[n], {n,20}] (* Harvey P. Dale, Aug 03 2011 *)
(* Next, the connection with Vandermonde determinants *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]   (* A203303 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *) (* Clark Kimberling, Jan 01 2011 *)
Table[(-1)^n*2^Binomial[n+1,2]*QPochhammer[2,2,n], {n,0,20}] (* G. C. Greubel, Aug 31 2023 *)

a(n)=if(n<0,0,prod(k=1,n,2^k-1)*2^((n^2+n)/2)) /* Michael Somos, May 09 2005 */

[product(2^(n+1) - 2^(k+1) for k in range(n)) for n in range(21)] # G. C. Greubel, Aug 31 2023

a(n) is asymptotic to C*2^(n*(n+1)) where C = 0.288788095086602421278899721... = prod(k>=1, 1-1/2^k) (cf. A048651). - Benoit Cloitre, Apr 11 2003
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)). [From Putman Exam]. - Max Alekseyev, May 18 2007
a(n) = 2*A203305(n), n > 0. - Clark Kimberling, Jan 01 2012
From Max Alekseyev, Jun 09 2015: (Start)
a(n) = 2^A000217(n) * A005329(n).
a(n) = 2^n * A002884(n).
a(n) = 2^n * n! * A053601(n). (End)
From G. C. Greubel, Aug 31 2023: (Start)
a(n) = Product_{j=0..n-1} (2^(n+1) - 2^(j+1)).
a(n) = (-1)^n * 2^binomial(n+1,2) * QPochhammer(2,2,n). (End)

A088389 Number of real regular n X n (0,1) matrices modulo rows permutations.

Original entry on oeis.org

1, 1, 3, 29, 940, 104286, 40050850, 53640013886, 251995529844792

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 08 2003

Antony Pearson, On Hidden Structures in Contaminated Symbolic Data, Ph. D. thesis, University of Colorado at Boulder (2020) 27835343.

Cf. A055165, A053601.

a(n) = A055165(n) / n!.

a(8) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

a(0)=1 prepended by Alois P. Heinz, Jun 18 2022

A270880 Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 28, 28, 0, 1, 400, 1680, 840, 0, 1, 10416, 168640, 277760, 83328, 0, 1, 525792, 36053248, 159989760, 139991040, 27998208, 0, 1, 51116992, 17811244032, 209056841728, 419919790080, 227569434624, 32509919232

Offset: 0

Michel Marcus, Mar 25 2016

			Triangle begins:
1;
0, 1;
0, 1, 3;
0, 1, 28, 28;
0, 1, 400, 1680, 840;
0, 1, 10416, 168640, 277760, 83328;
...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.

Cf. A053601 (right diagonal), A270881 (row sums), A270882.

g[n_] := q^Binomial[n, 2] *FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2;Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {m}]]], {m, 1, n}], {n, 1, 6}] // Grid (* Geoffrey Critzer, May 18 2017 *)

T(n,m) = Sum_ g(n)/(g(n_1)*g(n_2)***g(n_m))/(a_1!*a_2!***a_n!) where the sum is over all partitions of n into m parts and a_1,a_2,...,a_n is the part count signature of the partition and g(n) = A002884(n). - Geoffrey Critzer, May 18 2017 (after formula given in first Ellerman link above).

A377642 a(n) = (1/(n-1)!) * Product_{i=1..n-1} (2^n-2^i).

Original entry on oeis.org

1, 2, 12, 224, 13440, 2666496, 1791885312, 4161269661696, 33955960439439360, 987107315743488737280, 103404624282172311371513856, 39408968779516596852827017445376, 55084280201257118417007491904448757760, 284322478318511376197290687371005495020093440

Offset: 1

Nikita Babich, Nov 05 2024

Nikita Babich, Table of n, a(n) for n = 1..100

Appears to be main diagonal of A270882.

Cf. A000079, A000142, A002884, A028365, A048651, A053601.

Table[Product[2^n - 2^i, {i, 1, n - 1}]/Factorial[n - 1], {n, 1, 20}]

a(n)=prod(i=1, n-1, 2^n-2^i)/(n-1)! \\ Andrew Howroyd, Nov 10 2024

a(n) = (Product_{i=1..n-1}(2^n-2^i))/((n-1)!).
a(n) = A028365(n-1)/A000142(n-1).
a(n) = A000079(n-1) * A053601(n-1).
a(n) ~ A048651 * 2^(n*(n-1)) / (n-1)!. - Vaclav Kotesovec, Nov 13 2024

A088437 Number of n X n orthogonal matrices over GF(2) modulo permutations of rows.

Original entry on oeis.org

1, 1, 1, 2, 6, 32, 288, 4608, 130560, 6684672, 621674496, 106099113984, 33421220904960, 19556188689530880, 21359269286705627136, 43743783499173124374528, 168632285389312394463805440, 1227942828363775231508883701760, 16941927202935006869128068433182720, 444122456468619444070070837134825095168

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003

nonn

Also the number of distinct self-dual bases for GF(2^n) over GF(2). - Max Alekseyev, Feb 11 2008

Max Alekseyev, PARI scripts
Joerg Arndt, Matters Computational (The Fxtbook), see p. 910.
Dieter Jungnickel, Alfred J. Menezes and Scott A. Vanstone, On the Number of Self-Dual Bases of GF(q^m) Over GF(q), Proc. Amer. Math. Soc. 109 (1990), 23-29.

Cf. A053601, A135488.

/* based on http://home.gwu.edu/~maxal/gpscripts/nsdb.gp by Max Alekseyev */
sd(m,q) =
/* Number of distinct self-dual bases of GF(q^m) over GF(q) where q is a power of prime */
{
   if ( q%2 && !(m%2), return(0) );
   return ( (q%2 + 1) * prod(i=1,m-1, q^i - (i+1)%2) / m! );
}
vector(66, n, sd(n,2)) /* Joerg Arndt, Jul 03 2011 */

a(n) = A003053(n) / n!.

More terms from Max Alekseyev, Feb 11 2008

A258745 Order of general affine group AGL(n,2) (=A028365(n)) divided by (n+1).

Original entry on oeis.org

1, 1, 8, 336, 64512, 53329920, 184308203520, 2621599886868480, 152122702768688332800, 35820150273699719298416640, 34112245508649716682268134604800, 131089993748184007771243790830298726400, 2029650642403883210241235064170615545004032000

Offset: 0

Max Alekseyev, Jun 08 2015

nonn

Max Alekseyev, Table of n, a(n) for n = 0..100
Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, SZ Sequences: Binary-Based (0, 2^q)-Sequences, arXiv:2505.20434 [cs.GR], 2025. See pp. 7, 12.
Putnam Exam. 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.

a(n) = A028365(n) / (n+1) = 2^n * A002884(n) / (n+1) = 2^n * n! * A053601(n) / (n+1).

A298561 Triangle read by rows. T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.

Original entry on oeis.org

1, 3, 4, 28, 56, 57, 840, 2800, 2920, 2921, 83328, 499968, 539648, 540144, 540145, 27998208, 323534848, 363889408, 364556032, 364558048, 364558049, 32509919232, 765789208576, 904149876736, 906907414528, 906918338560, 906918346688, 906918346689

Offset: 1

Geoffrey Critzer, Jan 21 2018

			  1
  3,     4,
  28,    56,     57,
  840,   2800,   2920,   2921,
  83328, 499968, 539648, 540144, 540145,

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.

Cf. A270881 (main diagonal), A053601 (column 1), A298339.

nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2; Flatten[Table[Table[Transpose[
     Map[Drop[#, 1] &,Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}],z], {k, 1, nn}]]][[j, k]], {k, 1, j}], {j, 1, nn}]]

exp(Sum_{j=0...k} x^j/A002884(j)) = Sum_{n>=0} T(n,k)/A002884(n)*x^n.

A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

Original entry on oeis.org

1, 7, 7, 35, 105, 168, 155, 1085, 5208, 13888, 651, 9765, 109368, 874944, 3999744, 2667, 82677, 1984248, 37039296, 507967488, 4063739904, 10795, 680085, 33732216, 1349288640, 43177236480, 1036253675520, 14737830051840

Offset: 2

Geoffrey Critzer, Apr 28 2024

For n>=2 and 3<=k<=n, to construct a size k circuit of M[A]: Choose a basis b_1,b_2,...,b_{k-1} of a k-1 dimensional subspace of GF(2)^n. Append the vector b_1 + b_2 + ... + b_{k-1}.

			Triangle begins ...
    1;
    7,    7;
   35,   105,     168;
  155,  1085,    5208,    13888;
  651,  9765,  109368,   874944,   3999744;
 2667, 82677, 1984248, 37039296, 507967488, 4063739904;
...

J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.

Cf. A022166, A053601, A006095, A372350 (row sums).

nn = 8; Map[Select[#, # > 0 &] &, Table[Table[PadRight[Table[Product[(2^n - 2^i)/(2^k - 2^i), {i, 0, k - 1}], {k, 2, n}], nn], {n, 2, nn}][[All, j]]*    Table[Product[2^n - 2^i, {i, 0, n - 1}]/(n + 1)!, {n, 2, nn}][[j]], {j, 1, nn - 1}] // Transpose] // Grid

T(n,k) = A022166(n,k-1)*A053601(k-1)/k.
T(n,3) = A006095.
T(n,n+1) = A053601(n)/(n+1).

A053995 Bases of n-dimensional vector space over GF(3).

Original entry on oeis.org

1, 2, 24, 1872, 1010880, 3963053952, 116846682720768, 26600881786245651456, 47704425342545776582103040, 684471586202557775548246032261120, 79552147744537053441626504660608566263808, 75648987817302293231984664268437958161057494925312

Offset: 0

Vladeta Jovovic, Apr 05 2000

Cf. A053290, A053601.

a(n) = (3^n-1)(3^n-3)...(3^n-3^(n-1))/n!.
From Geoffrey Critzer, Sep 27 2022: (Start)
Sum_{n>=0} a(n)*x^n/A053290(n) = exp(x).
a(n) = A053290(n) / n!. (End)

A053996 Number of bases of n-dimensional vector space over GF(4).

Original entry on oeis.org

1, 3, 90, 30240, 123379200, 6462306385920, 4516376686991769600, 43295772825884473845350400, 5810971590951606258595918774272000, 11092372326294974332542866301794421571584000

Offset: 0

Vladeta Jovovic, Apr 05 2000

Cf. A053291, A053601.

a(n)=(4^n - 1)(4^n - 4)...(4^n - 4^(n-1))/n!

cp's OEIS Frontend

A028365 Order of general affine group over GF(2), AGL(n,2).

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A088389 Number of real regular n X n (0,1) matrices modulo rows permutations.

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A270880 Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

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A377642 a(n) = (1/(n-1)!) * Product_{i=1..n-1} (2^n-2^i).

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A088437 Number of n X n orthogonal matrices over GF(2) modulo permutations of rows.

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A258745 Order of general affine group AGL(n,2) (=A028365(n)) divided by (n+1).

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A298561 Triangle read by rows. T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.

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A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

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