%I M4415 #142 Aug 22 2024 18:03:48

%S 0,0,1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,

%T 44731051,178940587,715795115,2863245995,11453115051,45812722347,

%U 183251413675,733006703275,2932028910251,11728119835307,46912487729835

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

%C 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

%C 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

%C 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

%C 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

%D 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.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

%H T. D. Noe, <a href="/A006095/b006095.txt">Table of n, a(n) for n = 0..200</a>

%H L. Mariot and E. Formenti, <a href="http://openit.disco.unimib.it/~mariot/mf_counting_coprime_polynomials_2016.pdf">The number of coprime/non-coprime pairs of polynomials over F_2 with degree n and nonzero constant term</a>.

%H Ronald Orozco López, <a href="https://arxiv.org/abs/2408.08943">Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function</a>, arXiv:2408.08943 [math.CO], 2024. See p. 11.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H A. I. Solomon, C.-L. Ho and G. H. E. Duchamp, <a href="http://arxiv.org/abs/1205.4958">Degrees of entanglement for multipartite systems</a>, arXiv preprint arXiv:1205.4958 [quant-ph], 2012. - _N. J. A. Sloane_, Oct 23 2012

%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).

%F G.f.: x^2/((1-x)(1-2x)(1-4x)).

%F a(n) = (2^n - 1)*(2^(n-1) - 1)/3 = 4^n/6 - 2^(n-1) + 1/3.

%F Row sums of triangle A130324. - _Gary W. Adamson_, May 24 2007

%F a(n) = Stirling2(n+1,3) + Stirling2(n+1,4). - _Zerinvary Lajos_, Oct 04 2007; corrected by _R. J. Mathar_, Mar 19 2011

%F a(n) = A139250(2^(n-1) - 1), n >= 1. - _Omar E. Pol_, Mar 03 2011

%F a(n) = 4*a(n-1) + 2^(n-1) - 1, n >= 2. - _Vincenzo Librandi_, Mar 19 2011

%F 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

%F a(n) = Sum_{k=0..n-2} 2^k*C(2*n-k-2, k), n >= 2. - _Johannes W. Meijer_, Aug 19 2013

%F 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

%F a(n) = a(n-1) + A000217(A000225(n-1)), n > 0. - _Ivan N. Ianakiev_, Dec 11 2017

%F E.g.f.: (2*exp(x)-3*exp(2*x)+exp(4*x))/6. - _Paul Weisenhorn_, Aug 22 2021

%p a:= n-> add((4^(n-1-j) - 2^(n-1-j))/2, j=0..n-1):

%p seq(a(n), n=0..24); # _Zerinvary Lajos_, Jan 04 2007

%p 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]

%p a := n -> (2^n - 2)*(2^n - 1)/6:

%p seq(a(n), n = 0..24); # _Peter Luschny_, Jul 20 2021

%t 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 *)

%t 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 *)

%t (* Next, using elementary symmetric functions *)

%t f[k_] := 2^(k - 1); t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[2, t[n]]

%t Table[a[n], {n, 2, 32}] (* A203235 *)

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

%t (* _Clark Kimberling_, Dec 31 2011 *)

%t Table[QBinomial[n, 2, 2], {n, 0, 24}] (* _Arkadiusz Wesolowski_, Nov 12 2015 *)

%o (Sage) [gaussian_binomial(n,2,2) for n in range(0,25)] # _Zerinvary Lajos_, May 24 2009

%o (PARI) a(n) = (2^n - 1)*(2^(n-1) - 1)/3 \\ _Charles R Greathouse IV_, Jul 25 2011

%o (PARI) concat([0, 0], Vec(x^2/((1-x)*(1-2*x)*(1-4*x)) + O(x^50))) \\ _Altug Alkan_, Nov 12 2015

%Y First differences: A006516.

%Y Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016256, A023000, A023001, A075113, A130324, A203235.

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_