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A006095 Gaussian binomial coefficient [n,2] for q=2.
(Formerly M4415)
63
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

REFERENCES

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

L. Mariot, E. Formenti, The number of coprime/non-coprime pairs of polynomials over F_2 with degree n and nonzero constant term.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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)

FORMULA

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

MAPLE

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 := -1/(z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation.

MATHEMATICA

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 *)

PROG

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

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

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

CROSSREFS

First differences: A006516.

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

Sequence in context: A000588 A005285 * A171477 A265612 A005003 A243382

Adjacent sequences:  A006092 A006093 A006094 * A006096 A006097 A006098

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 19 08:36 EST 2018. Contains 299330 sequences. (Running on oeis4.)