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A006095
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Gaussian binomial coefficient [n,2] for q=2.
(Formerly M4415)
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29
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
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, 2012. - N. J. A. Sloane, Oct 23 2012
Index entries for linear recurrences with constant coefficients, signature (7,-14,8)
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FORMULA
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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) = sum{k=0..n-1, C(n+k-1,2k)*2^(n-k-1)} + 0^n/2. - Paul Barry, Oct 23 2009
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(2^k*C(2*n-k-2, k), k=0..n-2), n >= 2. - Johannes W. Meijer, Aug 19 2013
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MAPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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First differences: A006516. Cf. also A075113.
Cf. A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256.
Cf. A130324, A203235.
Sequence in context: A000588 A005285 * A171477 A005003 A243382 A242577
Adjacent sequences: A006092 A006093 A006094 * A006096 A006097 A006098
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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