A006095 -id:A006095 - cp's OEIS frontend

A006101 Gaussian binomial coefficient [ n,3 ] for q=3.

1, 40, 1210, 33880, 925771, 25095280, 678468820, 18326727760, 494894285941, 13362799477720, 360801469802830, 9741692640081640, 263026177881648511, 7101711092201899360, 191746238094034963240, 5177148775980218655520, 139783020078437440101481

Offset: 3

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=3..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 (40, -390, 1080, -729).

r:=3; q:=3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 06 2016

Table[QBinomial[n, 3, 3], {n, 3, 20}] (* Vincenzo Librandi, Nov 06 2016 *)

[gaussian_binomial(n,3,3) for n in range(3,17)] # Zerinvary Lajos, May 25 2009

G.f.: z^3/((1-z)(1-3z)(1-9z)(1-27z)). Simon Plouffe in his 1992 dissertation

a(n) = (27^n - 13*9^n + 39*3^n - 27)/11232. - Mitch Harris, Mar 23 2008

A203241 Second elementary symmetric function of the first n terms of (1,2,4,8,...).

Original entry on oeis.org

2, 14, 70, 310, 1302, 5334, 21590, 86870, 348502, 1396054, 5588310, 22361430, 89462102, 357881174, 1431590230, 5726491990, 22906230102, 91625444694, 366502827350, 1466013406550, 5864057820502, 23456239670614, 93824975459670

Offset: 2

Clark Kimberling, Dec 31 2011

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A006095.

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}]    (* A203241 *)
Table[a[n]/2, {n, 2, 32}]  (* A006095 *)

Vec(-2*x^2 / ((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Aug 15 2014

a(n) = 2*A006095(n).
From Colin Barker, Aug 15 2014: (Start)
a(n) = (2 - 3*2^n + 4^n)/3.
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: -2*x^2 / ((x-1)*(2*x-1)*(4*x-1)). (End)
a(n) = Sum_{k=0...n-2} 2^k*(2^(n-1)-1+2^k). - J. M. Bergot, Mar 21 2018

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

A097038 A Jacobsthal variant.

Original entry on oeis.org

0, 1, 1, 5, 7, 21, 35, 85, 155, 341, 651, 1365, 2667, 5461, 10795, 21845, 43435, 87381, 174251, 349525, 698027, 1398101, 2794155, 5592405, 11180715, 22369621, 44731051, 89478485, 178940587, 357913941, 715795115, 1431655765, 2863245995

Offset: 0

Paul Barry, Jul 19 2004

Convolution of A001045 and A077957.

Also interleaving of A002450(n+1) and A006095(n+1).

Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Cf. A001045, A077957.

Cf. A002450, A006095.

concat(0, Vec(x/((1-2*x^2)*(1-x-2*x^2)) + O(x^50))) \\ Michel Marcus, Nov 13 2015

vector(50, n, n--; 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2) \\ Altug Alkan, Nov 13 2015

G.f.: 1/(1-x-2*x^2) - 1/(1-2*x^2) = x/((1-2*x^2)*(1-x-2*x^2));
a(n) = 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2;
a(n) = sum{k=0..floor((n+1)/2), binomial(n-k+1, k-1)2^k };
a(n) = sum{k=0..n, 2^(k/2)(1+(-1)^k)A001045(n-k)/2 };
a(n) = A001045(n+1)-A077957(n).

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

A346463 a(n) = 6 * GaussBinomial(2n, 2, 2) / denominator(Bernoulli(2n, 1)).

Original entry on oeis.org

0, 1, 7, 93, 2159, 15841, 6141, 44731051, 8421119, 86113647, 3331843885, 127479517837, 103104368637, 750599904340651, 82824819807611, 80500035008073, 36170086393773823, 49191317521302203051, 2460603943675971, 12592977287514948283051, 89351501819019263845

Offset: 0

Peter Luschny, Jul 19 2021

nonn

Cf. A006095, A002445, A160014, A346464.

a := n -> (4^n - 2)*(4^n - 1) / mul(i, i=select(isprime, map(i->i+1, numtheory[divisors] (2*n)))): seq(a(n), n = 0..20);

Table[6 QBinomial[2 n, 2, 2] / Denominator[BernoulliB[2 n, 1]], {n, 0, 20}]

a(n) = (4^n - 2)*(4^n - 1)/Clausen(2*n, 1), where Clausen(n, k) = A160014(n, k).

A346464 a(n) = 6 * GaussBinomial(2n, 2, 2) Bernoulli(2*n, 1).

Original entry on oeis.org

0, 1, -7, 93, -2159, 79205, -4243431, 313117357, -30459187423, 3777547352949, -581776592603735, 108932905225448381, -24370170371013413967, 6419958293615735090053, -1967044830254804722091719, 693575524342402846796188365, -278846808098157253796358662591

Offset: 0

Peter Luschny, Jul 19 2021

sign

Cf. A006095, A000367, A002445, A346463.

a := n -> `if`(n = 0, 0, -2*n*(4^n - 2)*(4^n - 1)*Zeta(1 - 2*n)):
seq(a(n), n = 0..16);

Table[6 QBinomial[2 n, 2, 2] BernoulliB[2 n, 1], {n, 0, 16}]

a(n) = -2*n*(4^n - 2)*(4^n - 1)*zeta(1 - 2*n) for n >= 1.

A006099 Gaussian binomial coefficient [ n, n/2 ] for q=2.

Original entry on oeis.org

1, 1, 3, 7, 35, 155, 1395, 11811, 200787, 3309747, 109221651, 3548836819, 230674393235, 14877590196755, 1919209135381395, 246614610741341843, 63379954960524853651, 16256896431763117598611, 8339787869494479328087443, 4274137206973266943778085267

Offset: 0

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=0..50.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A065446.

Table[QBinomial[n,Floor[n/2],2],{n,0,20}] (* Harvey P. Dale, Sep 07 2013 *)

a(n) ~ c * 2^(n^2/4), where c = 1 / QPochhammer[1/2, 1/2] = A065446 = 3.46274661945506361153795734292443116454... if n is even, and c = 2^(-1/4) / QPochhammer[1/2, 1/2] = 2^(-1/4) * A065446 = 2.911811219231681420726836976930855961516... if n is odd. - Vaclav Kotesovec, Jun 22 2014

More terms from Harvey P. Dale, Sep 07 2013

A006102 Gaussian binomial coefficient [ n,4 ] for q=3.

Original entry on oeis.org

1, 121, 11011, 925771, 75913222, 6174066262, 500777836042, 40581331447162, 3287582741506063, 266307564861468823, 21571273555248777493, 1747282899667791058573, 141530177899268957392924, 11463951511551877750726204, 928580264181940191843785764, 75215006575885931519565302404

Offset: 4

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=4..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)

Partial sums of A226804. - Christian Krause, Dec 26 2022

r:=4; q:=3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

A006102:=-1/((z-1)*(81*z-1)*(3*z-1)*(9*z-1)*(27*z-1)); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Table[QBinomial[n, 4, 3], {n, 4, 24}] (* Vincenzo Librandi, Aug 02 2016 *)

[gaussian_binomial(n,4,3) for n in range(4,20)] # Zerinvary Lajos, May 25 2009

cp's OEIS Frontend

A006101 Gaussian binomial coefficient [ n,3 ] for q=3.

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A203241 Second elementary symmetric function of the first n terms of (1,2,4,8,...).

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

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

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A097038 A Jacobsthal variant.

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

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A346463 a(n) = 6 * GaussBinomial(2*n, 2, 2) / denominator(Bernoulli(2*n, 1)).

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A346463 a(n) = 6 * GaussBinomial(2n, 2, 2) / denominator(Bernoulli(2n, 1)).

A346464 a(n) = 6 * GaussBinomial(2n, 2, 2) Bernoulli(2*n, 1).