A022195 -id:A022195 - cp's OEIS frontend

A006095 Gaussian binomial coefficient [n, 2] for q = 2.

0, 0, 1, 7, 35, 155, 651, 2667, 10795, 43435, 174251, 698027, 2794155, 11180715, 44731051, 178940587, 715795115, 2863245995, 11453115051, 45812722347, 183251413675, 733006703275, 2932028910251, 11728119835307, 46912487729835

Offset: 0

N. J. A. Sloane

Number of 4-block coverings of an n-set where every element of the set is covered by exactly 3 blocks (if offset is 3), so a(n) = (1/4!)*(4^n-6*2^n+8). - Vladeta Jovovic, Feb 20 2001
Number of non-coprime pairs of polynomials (f,g) with binary coefficients where both f and g have degree n+1 and nonzero constant term. - Luca Mariot and Enrico Formenti, Sep 26 2016
Number of triplets found from the integers 1 to 2^n-1 by converting to binary and performing an XOR operation on the corresponding bits of each pair. Defining addition in this carryless way (0+0=1+1=0, 0+1=1+0=1), each triplet (A,B,C) has the property A+B=C, A+C=B and B+C=A. For example, n=3 gives the 7 triplets (1,2,3), (1,4,5), (1,6,7), (2,4,6), (2,5,7), (3,4,7) and (3,5,6). Each integer appears in the set of triplets 2^(n-1)-1 times, for example 3 for n=3. - Ian Duff, Oct 05 2019
Number of 2-dimensional vector subspaces of (Z_2)^n, so also number of Klein subgroups of the group (C_2)^n. - Robert FERREOL, Jul 28 2021

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.

T. D. Noe, Table of n, a(n) for n = 0..200
L. Mariot and E. Formenti, The number of coprime/non-coprime pairs of polynomials over F_2 with degree n and nonzero constant term.
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 11.
Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 6.
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
A. I. Solomon, C.-L. Ho and G. H. E. Duchamp, Degrees of entanglement for multipartite systems, arXiv preprint arXiv:1205.4958 [quant-ph], 2012. - _N. J. A. Sloane_, Oct 23 2012
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

First differences: A006516.
Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), this sequence (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).
Cf. A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016256, A023000, A023001, A075113, A130324, A203235.

a:= n-> add((4^(n-1-j) - 2^(n-1-j))/2, j=0..n-1):
seq(a(n), n=0..24); # Zerinvary Lajos, Jan 04 2007
A006095 := -z^2/(z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation. [adapted to offset 0 by Peter Luschny, Jul 20 2021]
a := n -> (2^n - 2)*(2^n - 1)/6:
seq(a(n), n = 0..24); # Peter Luschny, Jul 20 2021

Join[{a=0,b=0},Table[c=6*b-8*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)
CoefficientList[Series[x^2/((1-x)(1-2x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[{7,-14,8},{0,0,1},30] (* Harvey P. Dale, Jul 22 2011 *)
(* Next, using elementary symmetric functions *)
f[k_] := 2^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]    (* A203235 *)
Table[a[n]/2, {n, 2, 32}]  (* A006095 *)
(* Clark Kimberling, Dec 31 2011 *)
Table[QBinomial[n, 2, 2], {n, 0, 24}] (* Arkadiusz Wesolowski, Nov 12 2015 *)

a(n) = (2^n - 1)*(2^(n-1) - 1)/3 \\ Charles R Greathouse IV, Jul 25 2011

concat([0, 0], Vec(x^2/((1-x)*(1-2*x)*(1-4*x)) + O(x^50))) \\ Altug Alkan, Nov 12 2015

[gaussian_binomial(n,2,2) for n in range(0,25)] # Zerinvary Lajos, May 24 2009

G.f.: x^2/((1-x)(1-2x)(1-4x)).
a(n) = (2^n - 1)*(2^(n-1) - 1)/3 = 4^n/6 - 2^(n-1) + 1/3.
Row sums of triangle A130324. - Gary W. Adamson, May 24 2007
a(n) = Stirling2(n+1,3) + Stirling2(n+1,4). - Zerinvary Lajos, Oct 04 2007; corrected by R. J. Mathar, Mar 19 2011
a(n) = A139250(2^(n-1) - 1), n >= 1. - Omar E. Pol, Mar 03 2011
a(n) = 4*a(n-1) + 2^(n-1) - 1, n >= 2. - Vincenzo Librandi, Mar 19 2011
a(0) = 0, a(1) = 0, a(2) = 1, a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Harvey P. Dale, Jul 22 2011
a(n) = Sum_{k=0..n-2} 2^k*C(2*n-k-2, k), n >= 2. - Johannes W. Meijer, Aug 19 2013
a(n) = Sum_{i=0..n-2, j=i..n-2} 2^{i+j} = 2^0 * (2^0 + 2^1 + ... + 2^(n-2)) + 2^1 * (2^1 + 2^2 + ... + 2^(n-2)) + ... + 2^(n-2) * 2^(n-2), n>1. - J. M. Bergot, May 08 2017
a(n) = a(n-1) + A000217(A000225(n-1)), n > 0. - Ivan N. Ianakiev, Dec 11 2017
E.g.f.: (2*exp(x)-3*exp(2*x)+exp(4*x))/6. - Paul Weisenhorn, Aug 22 2021
From Peter Bala, Jul 01 2025: (Start)
G.f. assuming an offset of 0: exp( Sum_{n >= 1} b(3*n)/b(n)*x^n/n ) = 1 + 7*x + 35*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.
The following are examples of telescoping series:
Sum_{n >= 2} 2^n/a(n) = 6, follows from 1 - (1/6)*Sum_{k = 2..n} 2^k/a(k) = 1/(2^n - 1).
Sum_{n >= 2} 2^n/(a(n)*a(n+2)) = 6/49, follows from 1 - (49/6)*Sum_{k = 2..n} 2^k/(a(k)*a(k+2)) = 1/A006096(n+2);
Sum_{n >= 2} 4^n/(a(n)*a(n+2)) = 26/49, follows from 13 - (49/2)*Sum_{k = 2..n} 4^k/(a(k)*a(k+2)) = A086224(n)/A006096(n+2);
Sum_{n >= 2} 8^n/(a(n)*a(n+2)) = 129/49, follows from 43 - (49/3)*Sum_{k = 2..n} 8^k/(a(k)*a(k+2)) = A171479(n+1)/A006096(n+2). (End)

A006096 Gaussian binomial coefficient [n, 3] for q = 2.

Original entry on oeis.org

1, 15, 155, 1395, 11811, 97155, 788035, 6347715, 50955971, 408345795, 3269560515, 26167664835, 209386049731, 1675267338435, 13402854502595, 107225699266755, 857817047249091, 6862582190715075, 54900840777134275, 439207459223777475

Offset: 3

N. J. A. Sloane

42*a(n) is a maximum number of intercalates in a Latin square of order 2^n-1 (see A092237). - Eduard I. Vatutin, Apr 24 2025

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.

T. D. Noe, Table of n, a(n) for n=3..203
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 13.
Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 10.
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)
Eduard I. Vatutin, About interconnection between maximum number of intercalates in Latin squares of order N=2^n-1 and Gaussian binomial coefficients [n,3] for q=2 (in Russian).
Index entries for linear recurrences with constant coefficients, signature (15,-70,120,-64).

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

Cf. A092237.

r:=3; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2016

seq((-1+7*2^n-14*4^n+8*8^n)/21,n=1..20);
A006096:=1/(z-1)/(8*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Drop[CoefficientList[Series[x^3/((1 - x) (1 - 2 x) (1 - 4 x) (1 - 8 x)), {x, 0, 30}], x], 3]
QBinomial[Range[3,30],3,2] (* Harvey P. Dale, Jan 28 2013 *)

[gaussian_binomial(n,3,2) for n in range(3,23)] # Zerinvary Lajos, May 24 2009

G.f.: x^3/((1-x)(1-2x)(1-4x)(1-8x)).
(With a different offset) a(n) = (-1+7*2^n-14*4^n+8*8^n)/21. - James R. Buddenhagen, Dec 14 2003
From Peter Bala, Jul 01 2025: (Start)
a(n) = (q^n - 1)*(q^(n-1) - 1)*(q^(n-2) - 1)/((q^3 - 1)*(q^2 - 1)*(q - 1)) at q = 2.
G.f. with an offset of 0: exp( Sum_{n >= 1} b(4*n)/b(n)*x^n/n ) = 1 + 15*x + 155*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.
The following series telescope:
Sum_{n >= 3} 2^n/(a(n)*a(n+3)) = 420/72075;
Sum_{n >= 3} 4^n/(a(n)*a(n+3)) = 3416/72075;
Sum_{n >= 3} 8^n/(a(n)*a(n+3)) = 28296/72075;
Sum_{n >= 3} 16^n/(a(n)*a(n+3)) = 244748/72075;
Sum_{n >= 3} 32^n/(a(n)*a(n+3)) = 2415315/72075. (End)

A006097 Gaussian binomial coefficient [n, 4] for q = 2.

Original entry on oeis.org

1, 31, 651, 11811, 200787, 3309747, 53743987, 866251507, 13910980083, 222984027123, 3571013994483, 57162391576563, 914807651274739, 14638597687734259, 234230965858250739, 3747802679431278579, 59965700687947706355, 959458073589354016755, 15351384078270441402355

Offset: 4

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.

T. D. Noe, Table of n, a(n) for n = 4..204
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,-310,1240,-1984,1024).

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

r:=4; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2016

A006097:=-1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];
qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);
Table[qbin[n, 4, 2], {n, 4, 21}] (* Jean-François Alcover, Jul 21 2011 *)
QBinomial[Range[4,30],4,2] (* Harvey P. Dale, Dec 10 2012 *)

a(n)=(2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160 \\ Charles R Greathouse IV, Feb 19 2017

[gaussian_binomial(n,4,2) for n in range(4,22)] # Zerinvary Lajos, May 24 2009

G.f.: x^4/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)).
a(n) = (2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160. - Bruno Berselli, Aug 29 2011
From Peter Bala, Jul 01 2025: (Start)
G.f. with an offset of 0: exp( Sum_{n >= 1} b(5*n)/b(n)*x^n/n ) = 1 + 31*x + 651*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.
The following series telescope:
Sum_{n >= 4} 2^n/a(n) = 120/7; Sum_{n >= 4} 4^n/a(n) = 2078/7;
Sum_{n >= 4} 8^n/a(n) = 41280/7.
Sum_{n >= 4} 2^n/(a(n)*a(n+4)) = 40/499999;
Sum_{n >= 4} 2^n/(a(n)*a(n+4)*a(n+8)) = 40/6981154678721773;
Sum_{n >= 4} 2^n/(a(n)*a(n+4)*a(n+8)*a(n+12)) = 40/6387876185324781622646124392439. (End)

A006110 Gaussian binomial coefficient [n, 5] for q = 2.

Original entry on oeis.org

1, 63, 2667, 97155, 3309747, 109221651, 3548836819, 114429029715, 3675639930963, 117843461817939, 3774561792168531, 120843139740969555, 3867895279362300499, 123787287537281350227, 3961427432158861458003, 126769425631762997934675, 4056681585917103881615955, 129814770207420913565727315

Offset: 5

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 = 5..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 (63,-1302,11160,-41664,64512,-32768).

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

r:=5; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016

seq((1024*32^n-1984*16^n+1240*8^n-310*4^n+31*2^n-1)/9765,n=1..20);
A006110:=1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1)/(32*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Table[QBinomial[n, 5, 2], {n, 5, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

[gaussian_binomial(n,5,2) for n in range(5,18)] # Zerinvary Lajos, May 24 2009

a(n+4) = (1024*32^n-1984*16^n+1240*8^n-310*4^n+31*2^n-1)/9765. - James R. Buddenhagen, Dec 14 2003
G.f.: x^5/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)*(1-32*x)). - Vincenzo Librandi, Aug 07 2016
a(n) = Product_{i=1..5} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 06 2016
a(n) = (2^n-16)*(2^n-8)*(2^n-4)*(2^n-2)*(2^n-1)/9999360. - Robert Israel, Feb 01 2018
G.f. with an offset of 0: exp( Sum_{n >= 1} b(6*n)/b(n)*x^n/n ) = 1 + 63*x + 2667*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

A022189 Gaussian binomial coefficients [n, 6] for q = 2.

Original entry on oeis.org

1, 127, 10795, 788035, 53743987, 3548836819, 230674393235, 14877590196755, 955841412523283, 61291693863308051, 3926442969043883795, 251413193158549532435, 16094312257426532376339, 1030159771762835353435923

Offset: 6

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 6..200

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), this sequence (k = 6), A022190 - A022195 (k = 7 thru 12).

r:=6; 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, 6, 2], {n, 6, 24}] (* Vincenzo Librandi, Aug 03 2016 *)

r=6; 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,6,2) for n in range(6,20)] # Zerinvary Lajos, May 24 2009

a(n) = Product_{i=1..6} (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(7*n)/b(n)*x^n/n ) = 1 + 127*x + 10795*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Offset changed by Vincenzo Librandi, Aug 03 2016

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

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

cp's OEIS Frontend

A006095 Gaussian binomial coefficient [n, 2] for q = 2.

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A006096 Gaussian binomial coefficient [n, 3] for q = 2.

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A006097 Gaussian binomial coefficient [n, 4] for q = 2.

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A006110 Gaussian binomial coefficient [n, 5] for q = 2.

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A022189 Gaussian binomial coefficients [n, 6] for q = 2.

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

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