A006107 Gaussian binomial coefficient [ n,4 ] for q = 4.

1, 341, 93093, 24208613, 6221613541, 1594283908581, 408235958349285, 104514759495347685, 26756185103024942565, 6849609413493939400165, 1753501675591663698472421, 448896535558672700374937061, 114917519925881846404167134693

Offset: 4

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 = 4..200
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Programs

Magma

r:=4; q:=4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016

Maple

seq((4^n-64)*(4^n-16)*(4^n-4)*(4^n-1)/2961100800, n=4..30); # Robert Israel, Feb 01 2018

Mathematica

Table[QBinomial[n, 4, 4], {n, 4, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

Sage

[gaussian_binomial(n,4,4) for n in range(4,14)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^4/((1-x)*(1-4*x)*(1-16*x)*(1-64*x)*(1-256*x)). - Vincenzo Librandi, Aug 07 2016

a(n) = Product_{i=1..4} (4^(n-i+1)-1)/(4^i-1), by definition. - Vincenzo Librandi, Aug 07 2016

a(n) = (4^n-64)*(4^n-16)*(4^n-4)*(4^n-1)/2961100800. - Robert Israel, Feb 01 2018