|
| |
|
|
A006095
|
|
Gaussian binomial coefficient [n,2] for q=2.
(Formerly M4415)
|
|
26
| |
|
|
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; 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 (vladeta(AT)eunet.rs), Feb 20 2001
|
|
|
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
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to 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 (qntmpkt(AT)yahoo.com), May 24 2007
a(n)=stirling2(n+1,3)+stirling2(n+1,4). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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. [From Paul Barry (pbarry(AT)wit.ie), 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) [From Harvey P. Dale, Jul 22 2011]
|
|
|
MAPLE
| a:=n->sum((4^(n-j)-2^(n-j))/2, j=0..n): seq(a(n), n=-1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007
A006095:=-1/(z-1)/(2*z-1)/(4*z-1); [S. Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
| Join[{a=0, b=0}, Table[c=6*b-8*a+1; a=b; b=c, {n, 60}]] (*From 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] (* From 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 *)
|
|
|
PROG
| (Other) sage: [gaussian_binomial(n, 2, 2) for n in xrange(0, 25)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 24 2009]
(PARI) a(n) = (2^n - 1)*(2^(n-1) - 1)/3 \\ Charles R Greathouse IV, Jul 25 2011
|
|
|
CROSSREFS
| 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 A163348 A037099
Adjacent sequences: A006092 A006093 A006094 * A006096 A006097 A006098
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
