A006111 Gaussian binomial coefficient [ n,2 ] for q=5.

1, 31, 806, 20306, 508431, 12714681, 317886556, 7947261556, 198682027181, 4967053120931, 124176340230306, 3104408566792806, 77610214474995931, 1940255363400777181, 48506384092648824056, 1212659602354367574056, 30316490059049924214681

Offset: 2

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

• 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 (31,-155,125)

Programs

Maple

A006111:=-1/(z-1)/(25*z-1)/(5*z-1); # [Simon Plouffe in his 1992 dissertation with offset 0]

Mathematica

Transpose[NestList[Flatten[{Last[#],30Last[#]- 125First[#]+1}]&, {1,31}, 20]] [[1]]  (* Harvey P. Dale, Mar 26 2011 *)
LinearRecurrence[{31, -155, 125}, {1, 31, 806}, 10] (* T. D. Noe, Mar 26 2011 *)

Sage

[gaussian_binomial(n,2,5) for n in range(2,16)] # Zerinvary Lajos, May 28 2009

Formula

G.f.: x^2/[(1-x)(1-5x)(1-25x)].

a(n) = 6*a(n-1) - 5*a(n-2) + 25^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011

a(n) = 30*a(n-1) - 125*a(n-2) + 1, n>=3. - Vincenzo Librandi, Mar 20 2011

a(n) = -5^(n-1)/16 + 25^n/480 + 1/96. - R. J. Mathar, Mar 21 2011

Extensions

More terms from Harvey P. Dale, Mar 26 2011