A006096 -id:A006096 - cp's OEIS frontend

A034496 Sum of n-th powers of divisors of 8.

4, 15, 85, 585, 4369, 33825, 266305, 2113665, 16843009, 134480385, 1074791425, 8594130945, 68736258049, 549822930945, 4398314962945, 35185445863425, 281479271743489, 2251816993685505, 18014467229220865

Offset: 0

N. J. A. Sloane

Conjecture: No primes in this sequence (checked for first 10000 terms). [Artur Jasinski, Sep 23 2008]

All terms are composite because a(n) = (1 + 2^n)*(1 + 4^n). [T. D. Noe, Apr 26 2010]

T. D. Noe, Table of n, a(n) for n = 0..200
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.

Cf. A006096, A022166.

[DivisorSigma(n,8): n in [0..20]]; // Vincenzo Librandi, Apr 17 2014

Total[#^Range[0, 20]&/@Divisors[8]] (* Vincenzo Librandi, Apr 17 2014 *)
DivisorSigma[Range[0,20],8] (* Harvey P. Dale, May 16 2020 *)

a(n)=sigma(8, n) \\ Charles R Greathouse IV, May 16 2011

[sigma(8,n) for n in range(0,19)] # Zerinvary Lajos, Jun 04 2009

G.f.: (4 - 45*x + 140*x^2 - 120*x^3)/((1 - 8*x)*(1 - 4*x)*(1 - 2*x)*(1 - x)). [Bruno Berselli, Apr 17 2014]

a(n) = (2^(4*n) - 1)/( 2^n - 1) = 1 + 2^n + 4^n + 8^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 15*x + 155*x^2 + 1395*x^3 + ... is the o.g.f. for the 3rd subdiagonal of triangle A022166, essentially A006096. - Peter Bala, Apr 07 2015

A022190 Gaussian binomial coefficients [n, 7] for q = 2.

Original entry on oeis.org

1, 255, 43435, 6347715, 866251507, 114429029715, 14877590196755, 1919209135381395, 246614610741341843, 31627961868755063955, 4052305562169692070035, 518946525150879134496915, 66441249531569955747981459

Offset: 7

N. J. A. Sloane, Jun 14 1998

Vincenzo Librandi, Table of n, a(n) for n = 7..200

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

r:=7; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Table[QBinomial[n, 7, 2], {n, 7, 24}] (* Vincenzo Librandi, Aug 02 2016 *)

r=7; q=2; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, May 30 2018

[gaussian_binomial(n,7,2) for n in range(7,20)] # Zerinvary Lajos, May 25 2009

G.f.: x^7/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)*(1-32*x)*(1-64*x)*(1-128*x)). - Vincenzo Librandi, Aug 07 2016
a(n) = Product_{i=1..7} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 02 2016
G.f. with an offset of 0: exp( Sum_{n >= 1} b(8*n)/b(n)*x^n/n ) = 1 + 255*x + 43435*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Changed offset by Vincenzo Librandi, Aug 02 2016

A022191 Gaussian binomial coefficients [n, 8] for q = 2.

Original entry on oeis.org

1, 511, 174251, 50955971, 13910980083, 3675639930963, 955841412523283, 246614610741341843, 63379954960524853651, 16256896431763117598611, 4165817792093527797314451, 1066968301301093995246996371, 273210326382611632738979052435

Offset: 8

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 8..200

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

r:=8; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 03 2016

Table[QBinomial[n, 8, 2], {n, 8, 40}] (* Vincenzo Librandi, Aug 03 2016 *)

r=8; q=2; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, May 30 2018

[gaussian_binomial(n,8,2) for n in range(8,20)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..8} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 03 2016

G.f. with an offset of 0: exp( Sum_{n >= 1} b(9*n)/b(n)*x^n/n ) = 1 + 511*x +174251*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Offset changed by Vincenzo Librandi, Aug 03 2016

A022192 Gaussian binomial coefficients [n, 9] for q = 2.

Original entry on oeis.org

1, 1023, 698027, 408345795, 222984027123, 117843461817939, 61291693863308051, 31627961868755063955, 16256896431763117598611, 8339787869494479328087443, 4274137206973266943778085267, 2189425218271613769209626653075

Offset: 9

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 9..200

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

r:=9; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 03 2016

seq(eval(expand(QDifferenceEquations:-QBinomial(n,9,q)),q=2),n=9..50);

QBinomial[Range[9,20],9,2] (* Harvey P. Dale, Jul 24 2016 *)

r=9; q=2; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, May 30 2018

[gaussian_binomial(n,9,2) for n in range(9,21)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 02 2016
G.f.: x^9/Product_{0<=i<=9} (1-2^i*x). - Robert Israel, Apr 23 2017
G.f. with an offset of 0: exp( Sum_{n >= 1} b(10*n)/b(n)*x^n/n ) = 1 + 1023*x + 698027*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Offset changed by Vincenzo Librandi, Aug 03 2016

A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 13, 7, 1, 1, 31, 40, 35, 6, 1, 1, 63, 121, 155, 31, 12, 1, 1, 127, 364, 651, 156, 91, 8, 1, 1, 255, 1093, 2667, 781, 600, 57, 15, 1, 1, 511, 3280, 10795, 3906, 3751, 400, 155, 13, 1, 1, 1023, 9841, 43435, 19531, 22932, 2801, 1395, 130, 18, 1

Offset: 1

Ralf Stephan, May 09 2007

Differs from sum of divisors of m^(n-1) in 4th column!

			Array starts:
1,1,1,1,1,1,1,1,1,
1,3,4,7,6,12,8,15,13,
1,7,13,35,31,91,57,155,130,
1,15,40,155,156,600,400,1395,1210,
1,31,121,651,781,3751,2801,11811,11011,
1,63,364,2667,3906,22932,19608,97155,99463,
1,127,1093,10795,19531,138811,137257,788035,896260,
1,255,3280,43435,97656,836400,960800,6347715,8069620,

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

Álvar Ibeas, First 100 antidiagonals, flattened
Michael Baake, Solution of the coincidence problem in dimensions d≤4, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
B. Gruber, Alternative formulas for the number of sublattices, Acta Cryst. A53 (1997) 807-808.
Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.

Rows include A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.
Columns include A000225, A003462, A006095, A003463, A160869, A023000, A006096, A006100, A046915.
Transposed array is A160870.

T[n_, m_] := If[m == 1, 1, Product[{p, e} = pe; (p^(e+j)-1)/(p^j-1), {pe, FactorInteger[m]}, {j, 1, n-1}]];
Table[T[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 10 2018 *)

T(n,m)=local(k,v);v=factor(m);k=matsize(v)[1];prod(i=1,k,prod(j=1,n-1,(v[i,1]^(v[i,2]+j)-1)/(v[i,1]^j-1)))

Dirichlet g.f. of n-th row: Product_{i=0..n-1} zeta(s-i).
If m is squarefree, T(n,m) = A000203(m^(n-1)). - Álvar Ibeas, Jan 17 2015
T(n, Product(p^e)) = Product(Gaussian_poly[e+n-1, e]p). - _Álvar Ibeas, Oct 31 2015

Edited by Charles R Greathouse IV, Oct 28 2009

A022193 Gaussian binomial coefficients [n, 10] for q = 2.

Original entry on oeis.org

1, 2047, 2794155, 3269560515, 3571013994483, 3774561792168531, 3926442969043883795, 4052305562169692070035, 4165817792093527797314451, 4274137206973266943778085267, 4380990637147598617372537398675

Offset: 10

N. J. A. Sloane

nonn

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

Vincenzo Librandi, Table of n, a(n) for n = 10..200

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

r:=10; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 03 2016

Table[QBinomial[n, 10, 2], {n, 10, 40}] (* Vincenzo Librandi, Aug 03 2016 *)

r=10; q=2; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, May 30 2018

[gaussian_binomial(n,10,2) for n in range(10,21)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 03 2016

G.f. assuming an offset of 0: exp( Sum_{n >= 1} b(11*n)/b(n)*x^n/n ) = 1 + 2047*x + 2794155*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 03 2025

A022194 Gaussian binomial coefficients [n, 11] for q = 2.

Original entry on oeis.org

1, 4095, 11180715, 26167664835, 57162391576563, 120843139740969555, 251413193158549532435, 518946525150879134496915, 1066968301301093995246996371, 2189425218271613769209626653075, 4488323837657412597958687922896275

Offset: 11

N. J. A. Sloane

nonn

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

Vincenzo Librandi, Table of n, a(n) for n = 11..200

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

r:=11; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 03 2016

QBinomial[Range[11,30],11,2] (* Harvey P. Dale, Oct 21 2014 *)

r=11; q=2; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, May 30 2018

[gaussian_binomial(n,11,2) for n in range(11,22)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..11} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 03 2016

G.f. assuming an offset of 0: exp( Sum_{n >= 1} b(12*n)/b(n)*x^n/n ) = 1 + 4095*x + 11180715*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 03 2025

A288853 Triangle read by rows: T(n,k) is the number of surjective linear mappings from an n-dimensional vector space over F_2 onto a k-dimensional vector space, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 1, 7, 42, 168, 1, 15, 210, 2520, 20160, 1, 31, 930, 26040, 624960, 9999360, 1, 63, 3906, 234360, 13124160, 629959680, 20158709760, 1, 127, 16002, 1984248, 238109760, 26668293120, 2560156139520, 163849992929280, 1, 255, 64770, 16322040, 4047865920, 971487820800, 217613271859200, 41781748196966400, 5348063769211699200

Offset: 0

Geoffrey Critzer, Jun 18 2017

The (q = 2) analog of A008279.
A022166(m,k)*T(n,k) is the number of m X n matrices over F_2 that have rank k.
a(n) is the number of n X n matrices over F_2 in Green's R class containing A where rank(A) = k. - Geoffrey Critzer, Oct 05 2022

			  1;
  1,  1;
  1,  3,   6;
  1,  7,  42,   168;
  1, 15, 210,  2520,  20160;
  1, 31, 930, 26040, 624960, 9999360;
  ...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, 2010-2023, Example 2.3.6.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Wikipedia, Green's relations.

Columns k=0-10 give: A000012, A000225, 6*A006095, 168*A006096, 20160*A006097, 9999360*A006110, 20158709760*A022189, 163849992929280*A022190, 5348063769211699200*A022191, 699612310033197642547200*A022192, 366440137299948128422802227200*A022193.
Main diagonal gives A002884.
Cf. A022166.

Table[Table[Product[q^n - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0,8}] // Grid

T(n,k) = Product_{j=0..k-1} (2^n - 2^j).
T(n,k) = A002884(k)*A022166(n,k).
Let g_m(x) = Sum_{n>=0} (2^m*x)^n/A005329(n) and e(x) = Sum_{n>=0} x^n/A005329(n).  Then Sum_{k>=0} T(n,k)*x^k/A005329(k) = g_n(x)/e(x). - Geoffrey Critzer, Jun 01 2024

cp's OEIS Frontend

A034496 Sum of n-th powers of divisors of 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A022190 Gaussian binomial coefficients [n, 7] for q = 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A022191 Gaussian binomial coefficients [n, 8] for q = 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A022192 Gaussian binomial coefficients [n, 9] for q = 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A022193 Gaussian binomial coefficients [n, 10] for q = 2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage