A006097 Gaussian binomial coefficient [ n,4 ] for q=2. (Formerly M5226)

1, 31, 651, 11811, 200787, 3309747, 53743987, 866251507, 13910980083, 222984027123, 3571013994483, 57162391576563, 914807651274739, 14638597687734259, 234230965858250739, 3747802679431278579, 59965700687947706355, 959458073589354016755

Offset

4,2

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 = 4..204

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 (31,-310,1240,-1984,1024).

Formula

G.f.: x^4/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)).

a(n) = (2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160. - Bruno Berselli, Aug 29 2011

Maple

A006097:=-1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Mathematica

faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];
qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);
Table[qbin[n, 4, 2], {n, 4, 21}] (* Jean-François Alcover, Jul 21 2011 *)
QBinomial[Range[4, 30], 4, 2] (* Harvey P. Dale, Dec 10 2012 *)

Prog

(Sage) [gaussian_binomial(n, 4, 2) for n in range(4, 22)] # Zerinvary Lajos, May 24 2009
(Magma) r:=4; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2016
(PARI) a(n)=(2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160 \\ Charles R Greathouse IV, Feb 19 2017

Crossrefs

Sequence in context: A020983 A020981 A362512 * A000565 A014930 A196988

Adjacent sequences: A006094 A006095 A006096 * A006098 A006099 A006100

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved