A006113 Gaussian binomial coefficient [ n,4 ] for q = 5.
1, 781, 508431, 320327931, 200525284806, 125368356709806, 78360229974772306, 48975769621072897306, 30609934249224268600431, 19131218685276848401412931, 11957012900737114492991256681, 7473133215765585192791624069181, 4670708278954101902438990598678556
Offset: 4
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 = 4..200
- M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Programs
-
Magma
r:=4; q:=5; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016
-
Maple
qBinom := proc(n,m,q) mul( (1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ; end proc: A006113 := proc(n) qBinom(n,4,5) ; end proc: seq(A006113(n),n=4..16) ; # R. J. Mathar, Sep 28 2011 -
Mathematica
Table[QBinomial[n, 4, 5], {n, 4, 20}] (* Vincenzo Librandi, Aug 07 2016 *) -
Sage
[gaussian_binomial(n,4,5) for n in range(4,14)] # Zerinvary Lajos, May 27 2009
Formula
G.f.: x^4/((1-x)*(1-5*x)*(1-25*x)*(1-125*x)*(1-625*x)). - Vincenzo Librandi, Aug 07 2016
a(n) = Product_{i=1..4} (5^(n-i+1)-1)/(5^i-1), by definition. - Vincenzo Librandi, Aug 06 2016