A006110 Gaussian binomial coefficient [ n,5 ] for q = 2. (Formerly M5327)

1, 63, 2667, 97155, 3309747, 109221651, 3548836819, 114429029715, 3675639930963, 117843461817939, 3774561792168531, 120843139740969555, 3867895279362300499, 123787287537281350227, 3961427432158861458003, 126769425631762997934675, 4056681585917103881615955, 129814770207420913565727315

Offset

5,2

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

Vincenzo Librandi, Table of n, a(n) for n = 5..200

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 (63,-1302,11160,-41664,64512,-32768).

Formula

a(n+4) = (1024*32^n-1984*16^n+1240*8^n-310*4^n+31*2^n-1)/9765. - James R. Buddenhagen, Dec 14 2003

G.f.: x^5/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)*(1-32*x)). - Vincenzo Librandi, Aug 07 2016

a(n) = Product_{i=1..5} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 06 2016

a(n) = (2^n-16)*(2^n-8)*(2^n-4)*(2^n-2)*(2^n-1)/9999360. - Robert Israel, Feb 01 2018

Maple

seq((1024*32^n-1984*16^n+1240*8^n-310*4^n+31*2^n-1)/9765, n=1..20);
A006110:=1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1)/(32*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Mathematica

Table[QBinomial[n, 5, 2], {n, 5, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

Prog

(Sage) [gaussian_binomial(n, 5, 2) for n in range(5, 18)] # Zerinvary Lajos, May 24 2009
(Magma) r:=5; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016

Crossrefs

Cf. A006097.

Sequence in context: A075516 A004376 A094938 * A132051 A167987 A273437

Adjacent sequences: A006107 A006108 A006109 * A006111 A006112 A006113

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved