A006102 Gaussian binomial coefficient [ n,4 ] for q=3.

1, 121, 11011, 925771, 75913222, 6174066262, 500777836042, 40581331447162, 3287582741506063, 266307564861468823, 21571273555248777493, 1747282899667791058573, 141530177899268957392924, 11463951511551877750726204, 928580264181940191843785764, 75215006575885931519565302404

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

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..100
• 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)

Crossrefs

Partial sums of A226804. - Christian Krause, Dec 26 2022

Programs

Magma

r:=4; q:=3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Maple

A006102:=-1/((z-1)*(81*z-1)*(3*z-1)*(9*z-1)*(27*z-1)); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Mathematica

Table[QBinomial[n, 4, 3], {n, 4, 24}] (* Vincenzo Librandi, Aug 02 2016 *)

Sage

[gaussian_binomial(n,4,3) for n in range(4,20)] # Zerinvary Lajos, May 25 2009