A006111 Gaussian binomial coefficient [ n,2 ] for q=5. (Formerly M5228)

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

Offset

2,2

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

Table of n, a(n) for n=2..18.

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)

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

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 *)

Prog

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

Crossrefs

Sequence in context: A166488 A051587 A069380 * A183828 A180847 A202977

Adjacent sequences: A006108 A006109 A006110 * A006112 A006113 A006114

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Mar 26 2011

Status

approved