A006110 Gaussian binomial coefficient [n, 5] for q = 2.

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

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), this sequence (k = 5), A022189 - A022195 ( k = 6 thru 12).

Programs

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

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 *)

Sage

[gaussian_binomial(n,5,2) for n in range(5,18)] # Zerinvary Lajos, May 24 2009

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

G.f. with an offset of 0: exp( Sum_{n >= 1} b(6*n)/b(n)*x^n/n ) = 1 + 63*x + 2667*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025