A006095 -id:A006095 - cp's OEIS frontend

A006106 Gaussian binomial coefficient [ n,3 ] for q = 4.

1, 85, 5797, 376805, 24208613, 1550842085, 99277752549, 6354157930725, 406672215935205, 26027119554103525, 1665737215212030181, 106607206793565997285, 6822861635108183247077, 436663151052043168024805, 27946441769812674154891493

Offset: 3

N. J. A. Sloane

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (85, -1428, 5440, -4096).

r:=3; q:=4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016

Table[QBinomial[n, 3, 4], {n, 3, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

[gaussian_binomial(n,3,4) for n in range(3,15)] # Zerinvary Lajos, May 27 2009

G.f.: x^3/((1-x)*(1-4*x)*(1-16*x)*(1-64*x)). - Simon Plouffe in his 1992 dissertation

a(n) = Product_{i=1..3} (4^(n-i+1)-1)/(4^i-1), by definition. - Vincenzo Librandi, Aug 07 2016

A006111 Gaussian binomial coefficient [ n,2 ] for q=5.

Original entry on oeis.org

1, 31, 806, 20306, 508431, 12714681, 317886556, 7947261556, 198682027181, 4967053120931, 124176340230306, 3104408566792806, 77610214474995931, 1940255363400777181, 48506384092648824056, 1212659602354367574056, 30316490059049924214681

Offset: 2

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (31,-155,125)

A006111:=-1/(z-1)/(25*z-1)/(5*z-1); # [Simon Plouffe in his 1992 dissertation with offset 0]

Transpose[NestList[Flatten[{Last[#],30Last[#]- 125First[#]+1}]&, {1,31}, 20]] [[1]]  (* Harvey P. Dale, Mar 26 2011 *)
LinearRecurrence[{31, -155, 125}, {1, 31, 806}, 10] (* T. D. Noe, Mar 26 2011 *)

[gaussian_binomial(n,2,5) for n in range(2,16)] # Zerinvary Lajos, May 28 2009

G.f.: x^2/[(1-x)(1-5x)(1-25x)].
a(n) = 6*a(n-1) - 5*a(n-2) + 25^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011
a(n) = 30*a(n-1) - 125*a(n-2) + 1, n>=3. - Vincenzo Librandi, Mar 20 2011
a(n) = -5^(n-1)/16 + 25^n/480 + 1/96. - R. J. Mathar, Mar 21 2011

More terms from Harvey P. Dale, Mar 26 2011

A006112 Gaussian binomial coefficient [ n,3 ] for q = 5.

Original entry on oeis.org

1, 156, 20306, 2558556, 320327931, 40053706056, 5007031143556, 625886840206056, 78236053707784181, 9779511680526143556, 1222439084242108174806, 152804888634672088643556, 19100611156944225555440431, 2387576396558283557830831056

Offset: 3

N. J. A. Sloane

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (156, -4030, 19500, -15625).

r:=3; q:=5; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016

A006112:=1/(z-1)/(125*z-1)/(25*z-1)/(5*z-1); # [conjectured by Simon Plouffe in his 1992 dissertation]
seq((5^n-1)*(5^n-5)*(5^n-25)/1488000, n=3..30); # Robert Israel, Feb 01 2018

Table[QBinomial[n, 3, 5], {n, 3, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

[gaussian_binomial(n,3,5) for n in range(3,14)] # Zerinvary Lajos, May 27 2009

G.f.: x^3/((1-x)*(1-5*x)*(1-25*x)*(1-125*x)). - Vincenzo Librandi, Aug 07 2016
a(n) = Product_{i=1..3} (5^(n-i+1)-1)/(5^i-1), by definition. - Vincenzo Librandi, Aug 07 2016
a(n) = (5^n-1)*(5^n-5)*(5^n-25)/1488000. - Robert Israel, Feb 01 2018

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

A109765 Expansion of x/((4x-1)(2x-1)(x+1)).

Original entry on oeis.org

0, 1, 5, 23, 97, 399, 1617, 6511, 26129, 104687, 419089, 1677039, 6709521, 26840815, 107368721, 429485807, 1717965073, 6871903983, 27487703313, 109950988015, 439804301585, 1759217905391, 7036873019665, 28147494874863

Offset: 0

Creighton Dement, Aug 13 2005

In reference to the program code given, 1baseksumseq[C*D] = A001045 (Jacobsthal sequence, disregard signs).

Floretion Algebra Multiplication Program, FAMP Code: 1basejsumseq[C*D] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and D = + .5'i + .5'k - .5j' - .5k' + .5'ii' + .5'jj' + .5'jk' + .5'ki'; sumtype: sum[Y[15]] = sum[Y[ * ]], disregard signs

Index entries for linear recurrences with constant coefficients, signature (5,-2,-8).

Cf. A001045, A006516, A010036, A006095.

CoefficientList[Series[x/((4x-1)(2x-1)(x+1)),{x,0,30}],x] (* or *)
LinearRecurrence[{5,-2,-8},{0,1,5},30] (* Harvey P. Dale, Jan 02 2013 *)

a(n) = 5*a(n-1) - 2*a(n-2) - 8*a(n-3), n >= 3.
a(n) = (1/15)*(6*4^n-5*2^n-(-1)^n).
a(n+1) + a(n) = A006516(n+1).
a(n+2) - a(n) = A010036(n+1).

A260638 Irregular table: list of symmetric n X n matrices made from 2-binomial coefficients, read by rows, where the k-th row of any n X n matrix is filled with binomial coefficients [k-1,k-1]..[k+n-2,k-1] (for q=2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 7, 1, 7, 35, 1, 1, 1, 1, 1, 3, 7, 15, 1, 7, 35, 155, 1, 15, 155, 1395, 1, 1, 1, 1, 1, 1, 3, 7, 15, 31, 1, 7, 35, 155, 651, 1, 15, 155, 1395, 11811, 1, 31, 651, 11811, 200787, 1, 1, 1, 1, 1, 1, 1, 3, 7, 15, 31, 63, 1, 7, 35, 155

Offset: 1

Arkadiusz Wesolowski, Nov 11 2015

The determinant of the n X n matrix is 2^((n/6)*(2*n^2 - 3*n + 1)), that is, A185995(n-1).

The permanent is in A260639.

			The irregular table starts:
1;
1, 1;
1, 3;
1, 1, 1;
1, 3, 7;
1, 7, 35;

G. C. Greubel, Table of n, a(n) for n = 1..819

Cf. A000225, A006095, A185995, A260639.

Flatten@Flatten@Table[Table[QBinomial[r + c, r, 2], {r, 0, n}, {c, 0, n}], {n, 0, 5}]

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

A346295 a(n) = Sum_{k=0..n} (2^k + 1) * (2^k + 2) / 2.

Original entry on oeis.org

3, 9, 24, 69, 222, 783, 2928, 11313, 44466, 176307, 702132, 2802357, 11197110, 44763831, 179006136, 715926201, 2863508154, 11453639355, 45813770940, 183253510845, 733010897598, 2932037298879, 11728136612544, 46912521284289, 187650034805442, 750600038558403

Offset: 0

Paul Weisenhorn, Jul 13 2021

All terms are multiples of 3.

Roger B. Nelson, Proof without Words: A Triangular Sum, Mathematics Magazine Vol. 78, No. 5 (December 2005), p. 395.
Index entries for linear recurrences with constant coefficients, signature (8,-21,22,-8).

Cf. A028401 (first differences).

Cf. A006095, A076024.

a:= proc(n) option remember:
if n=0 then 3 else (2^n+1)*(2^n+2)/2+procname(n-1) fi:
end proc:
seq(a(n), n=0..30);

Accumulate @ Table[(2^k + 1)*(2^k + 2)/2, {k, 0, 25}] (* Amiram Eldar, Jul 27 2021 *)
LinearRecurrence[{8,-21,22,-8},{3,9,24,69},30] (* Harvey P. Dale, Nov 21 2021 *)

a(n)=sum(k=0, n, (2^k+1)*(2^k+2)/2); \\ Michel Marcus, Jul 16 2021

a(n) = (2^(n+1) + 4) * (2^(n+1) + 5) / 6 - 4 + n.
More generally: let f(n, b) be the triangular sum Sum_{k=0..n} (2^k+b) * (2^k+b+1) / 2.
f(n, b) = (2^(n+1) + 3*b + 1) * (2^(n+1) + 3*b + 2) / 6 - (b + 1)^2 + b*(b + 1)*n / 2.
G.f.: ((b^2+3*b+2)/2 - (3*b^2+8*b+4)*x + (4*b^2+8*b+3)*x^2) / ((4*x-1) * (2*x-1) * (x-1)^2).
E.g.f.: exp(x) * ((6*b+3)*exp(x) + 2*exp(3*x) + 3(b^2+b)*x/2 + (3*b^2-3*b-4) / 2) / 3.
Then b = -1 gives A006095, b = 0 gives A076024, b = 1 gives A346295, b = 2 gives A346375.
G.f.: 3*(5*x^2 - 5*x + 1) / ((4*x - 1) * (2*x - 1) * (x - 1)^2).
a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) for n > 3.
This recurrence is valid for all sequences f(n,b).
E.g.f.: exp(x) * (9*exp(x) + 2*exp(3*x) + 3*x - 2) / 3. - Stefano Spezia, Aug 13 2021

A346375 a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.

Original entry on oeis.org

6, 16, 37, 92, 263, 858, 3069, 11584, 44995, 177350, 704201, 2806476, 11205327, 44780242, 179038933, 715991768, 2863639259, 11453901534, 45814295265, 183254559460, 733012994791, 2932041493226, 11728145001197, 46912538061552, 187650068359923, 750600105667318, 3002400087124729

Offset: 0

Paul Weisenhorn, Jul 14 2021

Roger B. Nelson, Proof without Words: A Triangular Sum, Mathematics Magazine Vol. 78, No. 5 (December 2005), p. 395.
Index entries for linear recurrences with constant coefficients, signature (8,-21,22,-8).

Cf. A006095, A076024.

a:= proc(n) option remember:
     if n=0 then 6 else procname(n-1)+(2^n+3)*(2^n+2)/2 fi:
    end proc:
seq(a(n), n=0..26);

a[n_]:=Sum[(2^k+2)*(2^k+3)/2,{k,0,n}];Array[a,30,0] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

a(n) = sum(k=0, n, (2^k+2)*(2^k+3)/2); \\ Michel Marcus, Jul 28 2021

a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.
a(n) = (2^(n+1) + 7) * (2^(n+1) + 8)/6 - 9 + 3*n.
More generally: let f(n, b) = Sum_{k=0..n} (2^k + b) * (2^k + b + 1)/2 then f(n, b) = (2^(n+1) + 3*b + 1) * (2^(n+1) + 3*b + 2) / 6 - (b + 1)^2 + b*(b + 1)*n/2.
G.f.: ((b^2+3*b+2)/2 - (3*b^2+8*b+4)*x + (4*b^2+8*b+3)*x^2) / ((4*x-1) * (2*x-1) * (x-1)^2).
E.g.f.: exp(x)*((6*b+3)*exp(x) + 2*exp(3*x) + 3*(b^2+b)*x/2 +(3*b^2-3*b-4) / 2) / 3.
Then b = -1 gives A006095, b = 0 gives A076024, b = 1 gives A346295, b = 2 gives A346375.
a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) with n > 3.
This recurrence is valid for all sequences f(n, b).
G.f.: (35*x^2 - 32*x + 6) / ((4*x - 1) * (2*x - 1) * (x - 1)^2).
E.g.f.: exp(x) * (1 + 15*exp(x) + 2*exp(3*x) + 9*x)/3. - Stefano Spezia, Aug 15 2021

cp's OEIS Frontend

A006106 Gaussian binomial coefficient [ n,3 ] for q = 4.

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Magma

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Sage

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A006111 Gaussian binomial coefficient [ n,2 ] for q=5.

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Maple

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A006112 Gaussian binomial coefficient [ n,3 ] for q = 5.

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

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Magma

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PARI

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

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Magma

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PARI

Sage

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A109765 Expansion of x/((4*x-1)*(2*x-1)*(x+1)).

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A260638 Irregular table: list of symmetric n X n matrices made from 2-binomial coefficients, read by rows, where the k-th row of any n X n matrix is filled with binomial coefficients [k-1,k-1]..[k+n-2,k-1] (for q=2).

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A109765 Expansion of x/((4x-1)(2x-1)(x+1)).