A006095 -id:A006095 - cp's OEIS frontend

A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.

1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 357, 85, 1, 1, 341, 5797, 5797, 341, 1, 1, 1365, 93093, 376805, 93093, 1365, 1, 1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1, 1, 21845, 23859109, 1550842085, 6221613541

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157784(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    5,       1;
  1,   21,      21,        1;
  1,   85,     357,       85,        1;
  1,  341,    5797,     5797,      341,       1;
  1, 1365,   93093,   376805,    93093,    1365,    1;
  1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;

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

T. D. Noe, Rows n=0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for sequences related to Gaussian binomial coefficients

Cf. A006118 (row sums), A002450 (k=1), A006105 (k=2), A006106 (k=3).

A022168 := proc(n,m)
        A027637(n)/A027637(n-m)/A027637(m) ;
end proc: # R. J. Mathar, Nov 14 2011

gaussianBinom[n_, k_, q_] := Product[q^i - 1, {i, n}]/Product[q^j - 1, {j, n - k}]/Product[q^l - 1, {l, k}]; Column[Table[gaussianBinom[n, k, 4], {n, 0, 8}, {k, 0, n}], Center] (* Alonso del Arte, Nov 14 2011 *)
Table[QBinomial[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 4; T[n_, 0]:= 1; T[n_,n_]:= 1; T[n_,k_]:= T[n,k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1,k]]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{q=4; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 1. - Seiichi Manyama, May 09 2025

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

A022195 Gaussian binomial coefficients [n, 12] for q = 2.

Original entry on oeis.org

1, 8191, 44731051, 209386049731, 914807651274739, 3867895279362300499, 16094312257426532376339, 66441249531569955747981459, 273210326382611632738979052435, 1121258922081448861468067825426835, 4597164868683271949171164500871212435

Offset: 12

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 = 12..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 - A022194 (k = 6 thru 11).

r:=12; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Aug 03 2016

Table[QBinomial[n, 12, 2], {n, 12, 200}] (* Vincenzo Librandi, Aug 03 2016 *)

r=12; 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,12,2) for n in range(12,23)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..12} (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(13*n)/b(n)*x^n/n ) = 1 + 8191*x + 44731051*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 03 2025

Offset changed by Vincenzo Librandi, Aug 03 2016

A060487 Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4).

Original entry on oeis.org

1, 3, 1, 7, 57, 95, 43, 3, 35, 717, 3107, 4520, 2465, 445, 12, 155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70, 651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

			Triangle begins:
  [1, 3, 1];
  [7, 57, 95, 43, 3];
  [35, 717, 3107, 4520, 2465, 445, 12];
  [155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70];
  [651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465];
   ...
There are 205 tricoverings of a 4-set(cf. A060486): 7 4-block, 57 5-block, 95 6-block, 43 7-block and 3 8-block tricoverings.

Andrew Howroyd, Table of n, a(n) for n = 3..1157

Columns include A060483, A060484, A060485.
Row sums are A060486.
Cf. A006095, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
row(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])*y^(m-n)/(1+y))}
for(n=3, 8, print(Vecrev(row(3,n)))); \\ Andrew Howroyd, Dec 23 2018

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

A006100 Gaussian binomial coefficient [n, 2] for q = 3.

Original entry on oeis.org

1, 13, 130, 1210, 11011, 99463, 896260, 8069620, 72636421, 653757313, 5883904390, 52955405230, 476599444231, 4289397389563, 38604583680520, 347441274648040, 3126971536402441

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.

T. D. Noe, Table of n, a(n) for n=2..100
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 (13,-39,27).

Cf. A203243.

a:=n->sum((9^(n-j)-3^(n-j))/6,j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 15 2007
A006100:=-1/(z-1)/(3*z-1)/(9*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]    (* A203243 *)
Table[a[n]/3, {n, 2, 32}]  (* A006100 *)

[gaussian_binomial(n,2,3) for n in range(2,19)] # Zerinvary Lajos, May 25 2009

G.f.: x^2/[(1-x)(1-3x)(1-9x)].
a(n) = (9^n - 4*3^n + 3)/48. - Mitch Harris, Mar 23 2008
a(n) = 4*a(n-1) -3*a(n-2) +9^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011
From Peter Bala, Jul 01 2025: (Start)
G.f. with an offset of 0: exp(Sum_{n >= 1} b(3*n)/b(n)*x^n/n) = 1 + 13*x + 130*x^2 + ..., where b(n) = 3^n - 1.
The following series telescope:
Sum_{n >= 2} 3^n/a(n) = 12; Sum_{n >= 2} 3^n/(a(n)*a(n+3)) = 129/16900;
Sum_{n >= 2} 9^n/(a(n)*a(n+3)) = 1227/16900;
Sum_{n >= 2} 3^n/(a(n)*a(n+3)*a(n+6)) = 156706257/18829431219368770. (End)

A006105 Gaussian binomial coefficient [ n,2 ] for q=4.

Original entry on oeis.org

1, 21, 357, 5797, 93093, 1490853, 23859109, 381767589, 6108368805, 97734250405, 1563749404581, 25019996065701, 400319959420837, 6405119440211877, 102481911401303973

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.

Vincenzo Librandi, Table of n, a(n) for n = 2..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 (21,-84,64)

A006105:=-1/(z-1)/(4*z-1)/(16*z-1); # Simon Plouffe in his 1992 dissertation, assuming offset zero

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, 2, 4], {n, 2, 16}] (* Jean-François Alcover, Jul 21 2011 *)
CoefficientList[Series[1 / ((1 - x) (1 - 4 x) (1 - 16 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2013 *)
LinearRecurrence[{21,-84,64},{1,21,357},20] (* Harvey P. Dale, Feb 17 2020 *)

[gaussian_binomial(n,2,4) for n in range(2,17)] # Zerinvary Lajos, May 28 2009

G.f.: x^2/((1-x)*(1-4*x)*(1-16*x)). [Multiplied by x^2 to match offset by R. J. Mathar, Mar 11 2009]
a(n) = (16^n - 5*4^n + 4)/180. - Mitch Harris, Mar 23 2008
a(n) = 5*a(n-1) -4*a(n-2) +16^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011

A006117 Sum of Gaussian binomial coefficients [ n,k ] for q=3.

Original entry on oeis.org

1, 2, 6, 28, 212, 2664, 56632, 2052656, 127902864, 13721229088, 2544826627424, 815300788443072, 452436459318538048, 434188323928823259776, 722197777341507864283008, 2078153254879878944892861184, 10366904326991986000747424911616, 89478415088556766546699920236339712, 1338962661056423158371347974009398601216

Offset: 0

N. J. A. Sloane

			O.g.f.: A(x) = 1/(1-x) + x/((1-x)*(1-3x)) + x^2/((1-x)*(1-3x)*(1-9x)) + x^3/((1-x)*(1-3x)*(1-9x)*(1-27x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,2,6,28,212,2664,56632,...] = BINOMIAL([1,1,3,15,129,1833,43347,..]);
[1,3,15,129,1833,43347,1705623,...] = BINOMIAL^2([1,1,7,67,1081,...]);
[1,7,67,1081,29185,1277887,...] = BINOMIAL^6([1,1,19,415,12961,...]);
[1,19,415,12961,684361,58352707,...] = BINOMIAL^18([1,1,55,3187,...]);
[1,55,3187,219673,22634209,...] = BINOMIAL^54([1,1,163,27055,4805569,...]);
etc.
G.f. = 1 + 2*x + 6*x^2 + 28*x^3 + 212*x^4 + 2664*x^5 + 56632*x^6 + 2052656*x^7 + ...

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 = 0..90
R. Chapman et al., 2-modular lattices from ternary codes, J. Th. des Nombres de Bordeaux, 14 (2002), 73-85.
S. Hitzemann, W. Hochstattler, On the combinatorics of Galois numbers, Discr. Math. 310 (2010) 3551-3557, Galois Numbers G_{n}^(2).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

[n le 2 select n else 2*Self(n-1)+(3^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016

f:=n-> 1+ add( mul((3^(n-i)-1)/(3^(i+1)-1), i=0..k-1), k=1..n);

Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(3^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 3], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-3^j*x+x*O(x^n))), n) \\ Paul D. Hanna, Dec 06 2007

O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - 3^k*x). - Paul D. Hanna, Dec 06 2007
a(n) = 2*a(n-1)+(3^(n-1)-1)*a(n-2), n>1. [Hitzemann and Hochstattler] - R. J. Mathar, Aug 21 2013
a(n) ~ c * 3^(n^2/4), where c = EllipticTheta[3,0,1/3] / QPochhammer[1/3,1/3] = 3.019783845699... if n is even and c = EllipticTheta[2,0,1/3]/QPochhammer[1/3,1/3] = 3.018269046371... if n is odd. - Vaclav Kotesovec, Aug 21 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-6*a(n+1) + 3*a(n+2)) for all n in Z. - Michael Somos, Jan 25 2014

cp's OEIS Frontend

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